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&= f(t_n,\\ y_n) \\\\\nk_2 &= f\\bigl(t_n + \\tfrac{h}{2},\\ y_n + \\tfrac{h}{2}\\,k_1\\bigr) \\\\\nk_3 &= f\\bigl(t_n + \\tfrac{h}{2},\\ y_n + \\tfrac{h}{2}\\,k_2\\bigr) \\\\\nk_4 &= f\\bigl(t_n + h,\\ y_n + 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und das Verfahren wird zur Simpsonregel.",[564,2715,2716],{},"Eigenschaften:",[2718,2719,2720,2756,2835,2867],"ul",{},[2721,2722,2723,2726,2727,2755],"li",{},[568,2724,2725],{},"Konvergenzordnung 4",": Halbiert man ",[576,2728,2730,2743],{"className":2729},[579],[576,2731,2733],{"className":2732},[583],[585,2734,2735],{"xmlns":587},[589,2736,2737,2741],{},[592,2738,2739],{},[598,2740,914],{},[630,2742,914],{"encoding":632},[576,2744,2746],{"className":2745,"ariaHidden":622},[637],[576,2747,2749,2752],{"className":2748},[641],[576,2750],{"className":2751,"style":926},[645],[576,2753,914],{"className":2754},[650,654],", sinkt der globale Fehler etwa auf ein Sechzehntel.",[2721,2757,2758,2761,2762,2834],{},[568,2759,2760],{},"Explizit",": Jedes ",[576,2763,2765,2784],{"className":2764},[579],[576,2766,2768],{"className":2767},[583],[585,2769,2770],{"xmlns":587},[589,2771,2772,2781],{},[592,2773,2774],{},[757,2775,2776,2778],{},[598,2777,964],{},[598,2779,2780],{},"i",[630,2782,2783],{"encoding":632},"k_i",[576,2785,2787],{"className":2786,"ariaHidden":622},[637],[576,2788,2790,2793],{"className":2789},[641],[576,2791],{"className":2792,"style":2617},[645],[576,2794,2796,2799],{"className":2795},[650],[576,2797,964],{"className":2798,"style":1263},[650,654],[576,2800,2802],{"className":2801},[659],[576,2803,2805,2826],{"className":2804},[663,805],[576,2806,2808,2823],{"className":2807},[667],[576,2809,2812],{"className":2810,"style":2811},[671],"height:0.3117em;",[576,2813,2814,2817],{"style":1278},[576,2815],{"className":2816,"style":680},[679],[576,2818,2820],{"className":2819},[684,685,686,687],[576,2821,2780],{"className":2822},[650,654,687],[576,2824,829],{"className":2825},[828],[576,2827,2829],{"className":2828},[667],[576,2830,2832],{"className":2831,"style":836},[671],[576,2833],{}," ergibt sich aus bereits bekannten Werten.",[2721,2836,2837,2838,2866],{},"Pro Schritt sind vier Auswertungen von ",[576,2839,2841,2854],{"className":2840},[579],[576,2842,2844],{"className":2843},[583],[585,2845,2846],{"xmlns":587},[589,2847,2848,2852],{},[592,2849,2850],{},[598,2851,612],{},[630,2853,612],{"encoding":632},[576,2855,2857],{"className":2856,"ariaHidden":622},[637],[576,2858,2860,2863],{"className":2859},[641],[576,2861],{"className":2862,"style":2544},[645],[576,2864,612],{"className":2865,"style":716},[650,654]," nötig. Bei gleichem Aufwand ist RK4 meist viel genauer als das Euler-Verfahren.",[2721,2868,2869],{},"Bei steifen Problemen braucht auch RK4 sehr kleine Schritte; dort sind implizite Verfahren besser.",[559,2871,2873],{"id":2872},"beispiele","Beispiele",[564,2875,2876,739,2960,3027,3028,3089,3090,3514,3515,3577,3578,3618],{},[576,2877,2879,2901],{"className":2878},[579],[576,2880,2882],{"className":2881},[583],[585,2883,2884],{"xmlns":587},[589,2885,2886,2898],{},[592,2887,2888,2894,2896],{},[595,2889,2890,2892],{},[598,2891,600],{},[602,2893,606],{"mathvariant":604,"lspace":605,"rspace":605},[602,2895,609],{},[598,2897,600],{},[630,2899,2900],{"encoding":632},"y' = y",[576,2902,2904,2951],{"className":2903,"ariaHidden":622},[637],[576,2905,2907,2910,2942,2945,2948],{"className":2906},[641],[576,2908],{"className":2909,"style":646},[645],[576,2911,2913,2916],{"className":2912},[650],[576,2914,600],{"className":2915,"style":655},[650,654],[576,2917,2919],{"className":2918},[659],[576,2920,2922],{"className":2921},[663],[576,2923,2925],{"className":2924},[667],[576,2926,2928],{"className":2927,"style":672},[671],[576,2929,2930,2933],{"style":675},[576,2931],{"className":2932,"style":680},[679],[576,2934,2936],{"className":2935},[684,685,686,687],[576,2937,2939],{"className":2938},[650,687],[576,2940,606],{"className":2941},[650,687],[576,2943],{"className":2944,"style":698},[697],[576,2946,609],{"className":2947},[702],[576,2949],{"className":2950,"style":698},[697],[576,2952,2954,2957],{"className":2953},[641],[576,2955],{"className":2956,"style":857},[645],[576,2958,600],{"className":2959,"style":655},[650,654],[576,2961,2963,2987],{"className":2962},[579],[576,2964,2966],{"className":2965},[583],[585,2967,2968],{"xmlns":587},[589,2969,2970,2984],{},[592,2971,2972,2974,2976,2978,2980,2982],{},[598,2973,600],{},[602,2975,616],{"stretchy":615},[762,2977,764],{},[602,2979,628],{"stretchy":615},[602,2981,609],{},[762,2983,967],{},[630,2985,2986],{"encoding":632},"y(0) = 1",[576,2988,2990,3017],{"className":2989,"ariaHidden":622},[637],[576,2991,2993,2996,2999,3002,3005,3008,3011,3014],{"className":2992},[641],[576,2994],{"className":2995,"style":712},[645],[576,2997,600],{"className":2998,"style":655},[650,654],[576,3000,616],{"className":3001},[720],[576,3003,764],{"className":3004},[650],[576,3006,628],{"className":3007},[738],[576,3009],{"className":3010,"style":698},[697],[576,3012,609],{"className":3013},[702],[576,3015],{"className":3016,"style":698},[697],[576,3018,3020,3024],{"className":3019},[641],[576,3021],{"className":3022,"style":3023},[645],"height:0.6444em;",[576,3025,967],{"className":3026},[650],", exakte Lösung ",[576,3029,3031,3050],{"className":3030},[579],[576,3032,3034],{"className":3033},[583],[585,3035,3036],{"xmlns":587},[589,3037,3038,3047],{},[592,3039,3040],{},[595,3041,3042,3045],{},[598,3043,3044],{},"e",[598,3046,619],{},[630,3048,3049],{"encoding":632},"e^t",[576,3051,3053],{"className":3052,"ariaHidden":622},[637],[576,3054,3056,3060],{"className":3055},[641],[576,3057],{"className":3058,"style":3059},[645],"height:0.7936em;",[576,3061,3063,3066],{"className":3062},[650],[576,3064,3044],{"className":3065},[650,654],[576,3067,3069],{"className":3068},[659],[576,3070,3072],{"className":3071},[663],[576,3073,3075],{"className":3074},[667],[576,3076,3078],{"className":3077,"style":3059},[671],[576,3079,3080,3083],{"style":675},[576,3081],{"className":3082,"style":680},[679],[576,3084,3086],{"className":3085},[684,685,686,687],[576,3087,619],{"className":3088},[650,654,687],". Für diese Gleichung multipliziert ein Schritt mit ",[576,3091,3093,3149],{"className":3092},[579],[576,3094,3096],{"className":3095},[583],[585,3097,3098],{"xmlns":587},[589,3099,3100,3146],{},[592,3101,3102,3104,3106,3108,3110,3120,3122,3133,3135],{},[762,3103,967],{},[602,3105,1039],{},[598,3107,914],{},[602,3109,1039],{},[1043,3111,3112,3118],{},[595,3113,3114,3116],{},[598,3115,914],{},[762,3117,1015],{},[762,3119,1015],{},[602,3121,1039],{},[1043,3123,3124,3130],{},[595,3125,3126,3128],{},[598,3127,914],{},[762,3129,1092],{},[762,3131,3132],{},"6",[602,3134,1039],{},[1043,3136,3137,3143],{},[595,3138,3139,3141],{},[598,3140,914],{},[762,3142,1165],{},[762,3144,3145],{},"24",[630,3147,3148],{"encoding":632},"1 + h + \\frac{h^2}{2} + \\frac{h^3}{6} + \\frac{h^4}{24}",[576,3150,3152,3171,3190,3305,3414],{"className":3151,"ariaHidden":622},[637],[576,3153,3155,3159,3162,3165,3168],{"className":3154},[641],[576,3156],{"className":3157,"style":3158},[645],"height:0.7278em;vertical-align:-0.0833em;",[576,3160,967],{"className":3161},[650],[576,3163],{"className":3164,"style":1664},[697],[576,3166,1039],{"className":3167},[1668],[576,3169],{"className":3170,"style":1664},[697],[576,3172,3174,3178,3181,3184,3187],{"className":3173},[641],[576,3175],{"className":3176,"style":3177},[645],"height:0.7778em;vertical-align:-0.0833em;",[576,3179,914],{"className":3180},[650,654],[576,3182],{"className":3183,"style":1664},[697],[576,3185,1039],{"className":3186},[1668],[576,3188],{"className":3189,"style":1664},[697],[576,3191,3193,3197,3296,3299,3302],{"className":3192},[641],[576,3194],{"className":3195,"style":3196},[645],"height:1.3629em;vertical-align:-0.345em;",[576,3198,3200,3203,3293],{"className":3199},[650],[576,3201],{"className":3202},[720,1678],[576,3204,3206],{"className":3205},[1043],[576,3207,3209,3285],{"className":3208},[663,805],[576,3210,3212,3282],{"className":3211},[667],[576,3213,3216,3230,3238],{"className":3214,"style":3215},[671],"height:1.0179em;",[576,3217,3218,3221],{"style":1694},[576,3219],{"className":3220,"style":1253},[679],[576,3222,3224],{"className":3223},[684,685,686,687],[576,3225,3227],{"className":3226},[650,687],[576,3228,1015],{"className":3229},[650,687],[576,3231,3232,3235],{"style":1709},[576,3233],{"className":3234,"style":1253},[679],[576,3236],{"className":3237,"style":1717},[1716],[576,3239,3240,3243],{"style":1720},[576,3241],{"className":3242,"style":1253},[679],[576,3244,3246],{"className":3245},[684,685,686,687],[576,3247,3249],{"className":3248},[650,687],[576,3250,3252,3255],{"className":3251},[650,687],[576,3253,914],{"className":3254},[650,654,687],[576,3256,3258],{"className":3257},[659],[576,3259,3261],{"className":3260},[663],[576,3262,3264],{"className":3263},[667],[576,3265,3268],{"className":3266,"style":3267},[671],"height:0.8913em;",[576,3269,3271,3275],{"style":3270},"top:-2.931em;margin-right:0.0714em;",[576,3272],{"className":3273,"style":3274},[679],"height:2.5em;",[576,3276,3279],{"className":3277},[684,3278,1620,687],"reset-size3",[576,3280,1015],{"className":3281},[650,687],[576,3283,829],{"className":3284},[828],[576,3286,3288],{"className":3287},[667],[576,3289,3291],{"className":3290,"style":1742},[671],[576,3292],{},[576,3294],{"className":3295},[738,1678],[576,3297],{"className":3298,"style":1664},[697],[576,3300,1039],{"className":3301},[1668],[576,3303],{"className":3304,"style":1664},[697],[576,3306,3308,3311,3405,3408,3411],{"className":3307},[641],[576,3309],{"className":3310,"style":3196},[645],[576,3312,3314,3317,3402],{"className":3313},[650],[576,3315],{"className":3316},[720,1678],[576,3318,3320],{"className":3319},[1043],[576,3321,3323,3394],{"className":3322},[663,805],[576,3324,3326,3391],{"className":3325},[667],[576,3327,3329,3343,3351],{"className":3328,"style":3215},[671],[576,3330,3331,3334],{"style":1694},[576,3332],{"className":3333,"style":1253},[679],[576,3335,3337],{"className":3336},[684,685,686,687],[576,3338,3340],{"className":3339},[650,687],[576,3341,3132],{"className":3342},[650,687],[576,3344,3345,3348],{"style":1709},[576,3346],{"className":3347,"style":1253},[679],[576,3349],{"className":3350,"style":1717},[1716],[576,3352,3353,3356],{"style":1720},[576,3354],{"className":3355,"style":1253},[679],[576,3357,3359],{"className":3358},[684,685,686,687],[576,3360,3362],{"className":3361},[650,687],[576,3363,3365,3368],{"className":3364},[650,687],[576,3366,914],{"className":3367},[650,654,687],[576,3369,3371],{"className":3370},[659],[576,3372,3374],{"className":3373},[663],[576,3375,3377],{"className":3376},[667],[576,3378,3380],{"className":3379,"style":3267},[671],[576,3381,3382,3385],{"style":3270},[576,3383],{"className":3384,"style":3274},[679],[576,3386,3388],{"className":3387},[684,3278,1620,687],[576,3389,1092],{"className":3390},[650,687],[576,3392,829],{"className":3393},[828],[576,3395,3397],{"className":3396},[667],[576,3398,3400],{"className":3399,"style":1742},[671],[576,3401],{},[576,3403],{"className":3404},[738,1678],[576,3406],{"className":3407,"style":1664},[697],[576,3409,1039],{"className":3410},[1668],[576,3412],{"className":3413,"style":1664},[697],[576,3415,3417,3420],{"className":3416},[641],[576,3418],{"className":3419,"style":3196},[645],[576,3421,3423,3426,3511],{"className":3422},[650],[576,3424],{"className":3425},[720,1678],[576,3427,3429],{"className":3428},[1043],[576,3430,3432,3503],{"className":3431},[663,805],[576,3433,3435,3500],{"className":3434},[667],[576,3436,3438,3452,3460],{"className":3437,"style":3215},[671],[576,3439,3440,3443],{"style":1694},[576,3441],{"className":3442,"style":1253},[679],[576,3444,3446],{"className":3445},[684,685,686,687],[576,3447,3449],{"className":3448},[650,687],[576,3450,3145],{"className":3451},[650,687],[576,3453,3454,3457],{"style":1709},[576,3455],{"className":3456,"style":1253},[679],[576,3458],{"className":3459,"style":1717},[1716],[576,3461,3462,3465],{"style":1720},[576,3463],{"className":3464,"style":1253},[679],[576,3466,3468],{"className":3467},[684,685,686,687],[576,3469,3471],{"className":3470},[650,687],[576,3472,3474,3477],{"className":3473},[650,687],[576,3475,914],{"className":3476},[650,654,687],[576,3478,3480],{"className":3479},[659],[576,3481,3483],{"className":3482},[663],[576,3484,3486],{"className":3485},[667],[576,3487,3489],{"className":3488,"style":3267},[671],[576,3490,3491,3494],{"style":3270},[576,3492],{"className":3493,"style":3274},[679],[576,3495,3497],{"className":3496},[684,3278,1620,687],[576,3498,1165],{"className":3499},[650,687],[576,3501,829],{"className":3502},[828],[576,3504,3506],{"className":3505},[667],[576,3507,3509],{"className":3508,"style":1742},[671],[576,3510],{},[576,3512],{"className":3513},[738,1678],", den ersten Gliedern der Exponentialreihe. Mit ",[576,3516,3518,3537],{"className":3517},[579],[576,3519,3521],{"className":3520},[583],[585,3522,3523],{"xmlns":587},[589,3524,3525,3534],{},[592,3526,3527,3529,3531],{},[598,3528,914],{},[602,3530,609],{},[762,3532,3533],{},"0,5",[630,3535,3536],{"encoding":632},"h = 0{,}5",[576,3538,3540,3558],{"className":3539,"ariaHidden":622},[637],[576,3541,3543,3546,3549,3552,3555],{"className":3542},[641],[576,3544],{"className":3545,"style":926},[645],[576,3547,914],{"className":3548},[650,654],[576,3550],{"className":3551,"style":698},[697],[576,3553,609],{"className":3554},[702],[576,3556],{"className":3557,"style":698},[697],[576,3559,3561,3564,3567,3573],{"className":3560},[641],[576,3562],{"className":3563,"style":2487},[645],[576,3565,764],{"className":3566},[650],[576,3568,3570],{"className":3569},[650],[576,3571,623],{"className":3572},[727],[576,3574,3576],{"className":3575},[650],"5"," ist der Faktor ",[576,3579,3581,3596],{"className":3580},[579],[576,3582,3584],{"className":3583},[583],[585,3585,3586],{"xmlns":587},[589,3587,3588,3593],{},[592,3589,3590],{},[762,3591,3592],{},"1,6484375",[630,3594,3595],{"encoding":632},"1{,}6484375",[576,3597,3599],{"className":3598,"ariaHidden":622},[637],[576,3600,3602,3605,3608,3614],{"className":3601},[641],[576,3603],{"className":3604,"style":2487},[645],[576,3606,967],{"className":3607},[650],[576,3609,3611],{"className":3610},[650],[576,3612,623],{"className":3613},[727],[576,3615,3617],{"className":3616},[650],"6484375",", nach zwei Schritten",[576,3620,3622],{"className":3621},[934],[576,3623,3625,3669],{"className":3624},[579],[576,3626,3628],{"className":3627},[583],[585,3629,3630],{"xmlns":587,"display":943},[589,3631,3632,3666],{},[592,3633,3634,3640,3642,3648,3651,3654,3656,3659,3661,3663],{},[757,3635,3636,3638],{},[598,3637,600],{},[762,3639,1015],{},[602,3641,609],{},[595,3643,3644,3646],{},[762,3645,3592],{},[762,3647,1015],{},[602,3649,3650],{},"≈",[762,3652,3653],{},"2,7173",[602,3655,623],{"separator":622},[697,3657],{"width":3658},"2em",[598,3660,3044],{},[602,3662,3650],{},[762,3664,3665],{},"2,7183",[630,3667,3668],{"encoding":632},"y_2 = 1{,}6484375^2 \\approx 2{,}7173, \\qquad e \\approx 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Fehler ist etwa ",[576,3851,3853,3868],{"className":3852},[579],[576,3854,3856],{"className":3855},[583],[585,3857,3858],{"xmlns":587},[589,3859,3860,3865],{},[592,3861,3862],{},[762,3863,3864],{},"0,00094",[630,3866,3867],{"encoding":632},"0{,}00094",[576,3869,3871],{"className":3870,"ariaHidden":622},[637],[576,3872,3874,3877,3880,3886],{"className":3873},[641],[576,3875],{"className":3876,"style":2487},[645],[576,3878,764],{"className":3879},[650],[576,3881,3883],{"className":3882},[650],[576,3884,623],{"className":3885},[727],[576,3887,3889],{"className":3888},[650],"00094",". Das Euler-Verfahren mit derselben Zahl von acht Auswertungen (",[576,3892,3894,3913],{"className":3893},[579],[576,3895,3897],{"className":3896},[583],[585,3898,3899],{"xmlns":587},[589,3900,3901,3910],{},[592,3902,3903,3905,3907],{},[598,3904,914],{},[602,3906,609],{},[762,3908,3909],{},"0,125",[630,3911,3912],{"encoding":632},"h = 0{,}125",[576,3914,3916,3934],{"className":3915,"ariaHidden":622},[637],[576,3917,3919,3922,3925,3928,3931],{"className":3918},[641],[576,3920],{"className":3921,"style":926},[645],[576,3923,914],{"className":3924},[650,654],[576,3926],{"className":3927,"style":698},[697],[576,3929,609],{"className":3930},[702],[576,3932],{"className":3933,"style":698},[697],[576,3935,3937,3940,3943,3949],{"className":3936},[641],[576,3938],{"className":3939,"style":2487},[645],[576,3941,764],{"className":3942},[650],[576,3944,3946],{"className":3945},[650],[576,3947,623],{"className":3948},[727],[576,3950,3952],{"className":3951},[650],"125",") liefert ",[576,3955,3957,3982],{"className":3956},[579],[576,3958,3960],{"className":3959},[583],[585,3961,3962],{"xmlns":587},[589,3963,3964,3979],{},[592,3965,3966,3974,3976],{},[595,3967,3968,3971],{},[762,3969,3970],{},"1,125",[762,3972,3973],{},"8",[602,3975,3650],{},[762,3977,3978],{},"2,5658",[630,3980,3981],{"encoding":632},"1{,}125^8 \\approx 2{,}5658",[576,3983,3985,4044],{"className":3984,"ariaHidden":622},[637],[576,3986,3988,3992,3995,4001,4005,4035,4038,4041],{"className":3987},[641],[576,3989],{"className":3990,"style":3991},[645],"height:1.0085em;vertical-align:-0.1944em;",[576,3993,967],{"className":3994},[650],[576,3996,3998],{"className":3997},[650],[576,3999,623],{"className":4000},[727],[576,4002,4004],{"className":4003},[650],"12",[576,4006,4008,4011],{"className":4007},[650],[576,4009,3576],{"className":4010},[650],[576,4012,4014],{"className":4013},[659],[576,4015,4017],{"className":4016},[663],[576,4018,4020],{"className":4019},[667],[576,4021,4024],{"className":4022,"style":4023},[671],"height:0.8141em;",[576,4025,4026,4029],{"style":675},[576,4027],{"className":4028,"style":680},[679],[576,4030,4032],{"className":4031},[684,685,686,687],[576,4033,3973],{"className":4034},[650,687],[576,4036],{"className":4037,"style":698},[697],[576,4039,3650],{"className":4040},[702],[576,4042],{"className":4043,"style":698},[697],[576,4045,4047,4050,4053,4059],{"className":4046},[641],[576,4048],{"className":4049,"style":2487},[645],[576,4051,1015],{"className":4052},[650],[576,4054,4056],{"className":4055},[650],[576,4057,623],{"className":4058},[727],[576,4060,4062],{"className":4061},[650],"5658",", also einen Fehler von ",[576,4065,4067,4082],{"className":4066},[579],[576,4068,4070],{"className":4069},[583],[585,4071,4072],{"xmlns":587},[589,4073,4074,4079],{},[592,4075,4076],{},[762,4077,4078],{},"0,1525",[630,4080,4081],{"encoding":632},"0{,}1525",[576,4083,4085],{"className":4084,"ariaHidden":622},[637],[576,4086,4088,4091,4094,4100],{"className":4087},[641],[576,4089],{"className":4090,"style":2487},[645],[576,4092,764],{"className":4093},[650],[576,4095,4097],{"className":4096},[650],[576,4098,623],{"className":4099},[727],[576,4101,4103],{"className":4102},[650],"1525",".",[564,4106,4107,739,4214,4279,4280,4339],{},[576,4108,4110,4136],{"className":4109},[579],[576,4111,4113],{"className":4112},[583],[585,4114,4115],{"xmlns":587},[589,4116,4117,4133],{},[592,4118,4119,4125,4127,4129,4131],{},[595,4120,4121,4123],{},[598,4122,600],{},[602,4124,606],{"mathvariant":604,"lspace":605,"rspace":605},[602,4126,609],{},[598,4128,619],{},[602,4130,1039],{},[598,4132,600],{},[630,4134,4135],{"encoding":632},"y' = t + y",[576,4137,4139,4186,4205],{"className":4138,"ariaHidden":622},[637],[576,4140,4142,4145,4177,4180,4183],{"className":4141},[641],[576,4143],{"className":4144,"style":646},[645],[576,4146,4148,4151],{"className":4147},[650],[576,4149,600],{"className":4150,"style":655},[650,654],[576,4152,4154],{"className":4153},[659],[576,4155,4157],{"className":4156},[663],[576,4158,4160],{"className":4159},[667],[576,4161,4163],{"className":4162,"style":672},[671],[576,4164,4165,4168],{"style":675},[576,4166],{"className":4167,"style":680},[679],[576,4169,4171],{"className":4170},[684,685,686,687],[576,4172,4174],{"className":4173},[650,687],[576,4175,606],{"className":4176},[650,687],[576,4178],{"className":4179,"style":698},[697],[576,4181,609],{"className":4182},[702],[576,4184],{"className":4185,"style":698},[697],[576,4187,4189,4193,4196,4199,4202],{"className":4188},[641],[576,4190],{"className":4191,"style":4192},[645],"height:0.6984em;vertical-align:-0.0833em;",[576,4194,619],{"className":4195},[650,654],[576,4197],{"className":4198,"style":1664},[697],[576,4200,1039],{"className":4201},[1668],[576,4203],{"className":4204,"style":1664},[697],[576,4206,4208,4211],{"className":4207},[641],[576,4209],{"className":4210,"style":857},[645],[576,4212,600],{"className":4213,"style":655},[650,654],[576,4215,4217,4240],{"className":4216},[579],[576,4218,4220],{"className":4219},[583],[585,4221,4222],{"xmlns":587},[589,4223,4224,4238],{},[592,4225,4226,4228,4230,4232,4234,4236],{},[598,4227,600],{},[602,4229,616],{"stretchy":615},[762,4231,764],{},[602,4233,628],{"stretchy":615},[602,4235,609],{},[762,4237,967],{},[630,4239,2986],{"encoding":632},[576,4241,4243,4270],{"className":4242,"ariaHidden":622},[637],[576,4244,4246,4249,4252,4255,4258,4261,4264,4267],{"className":4245},[641],[576,4247],{"className":4248,"style":712},[645],[576,4250,600],{"className":4251,"style":655},[650,654],[576,4253,616],{"className":4254},[720],[576,4256,764],{"className":4257},[650],[576,4259,628],{"className":4260},[738],[576,4262],{"className":4263,"style":698},[697],[576,4265,609],{"className":4266},[702],[576,4268],{"className":4269,"style":698},[697],[576,4271,4273,4276],{"className":4272},[641],[576,4274],{"className":4275,"style":3023},[645],[576,4277,967],{"className":4278},[650],", ein Schritt mit ",[576,4281,4283,4300],{"className":4282},[579],[576,4284,4286],{"className":4285},[583],[585,4287,4288],{"xmlns":587},[589,4289,4290,4298],{},[592,4291,4292,4294,4296],{},[598,4293,914],{},[602,4295,609],{},[762,4297,3533],{},[630,4299,3536],{"encoding":632},[576,4301,4303,4321],{"className":4302,"ariaHidden":622},[637],[576,4304,4306,4309,4312,4315,4318],{"className":4305},[641],[576,4307],{"className":4308,"style":926},[645],[576,4310,914],{"className":4311},[650,654],[576,4313],{"className":4314,"style":698},[697],[576,4316,609],{"className":4317},[702],[576,4319],{"className":4320,"style":698},[697],[576,4322,4324,4327,4330,4336],{"className":4323},[641],[576,4325],{"className":4326,"style":2487},[645],[576,4328,764],{"className":4329},[650],[576,4331,4333],{"className":4332},[650],[576,4334,623],{"className":4335},[727],[576,4337,3576],{"className":4338},[650],":",[576,4341,4343],{"className":4342},[934],[576,4344,4346,4442],{"className":4345},[579],[576,4347,4349],{"className":4348},[583],[585,4350,4351],{"xmlns":587,"display":943},[589,4352,4353,4439],{},[592,4354,4355,4361,4363,4365,4367,4370,4376,4378,4381,4383,4386,4388,4391,4393,4395,4401,4403,4405,4407,4410,4412,4415,4417,4419,4425,4427,4429,4431,4434,4436],{},[757,4356,4357,4359],{},[598,4358,964],{},[762,4360,967],{},[602,4362,609],{},[762,4364,967],{},[602,4366,623],{"separator":622},[697,4368],{"width":4369},"1em",[757,4371,4372,4374],{},[598,4373,964],{},[762,4375,1015],{},[602,4377,609],{},[762,4379,4380],{},"0,25",[602,4382,1039],{},[762,4384,4385],{},"1,25",[602,4387,609],{},[762,4389,4390],{},"1,5",[602,4392,623],{"separator":622},[697,4394],{"width":4369},[757,4396,4397,4399],{},[598,4398,964],{},[762,4400,1092],{},[602,4402,609],{},[762,4404,4380],{},[602,4406,1039],{},[762,4408,4409],{},"1,375",[602,4411,609],{},[762,4413,4414],{},"1,625",[602,4416,623],{"separator":622},[697,4418],{"width":4369},[757,4420,4421,4423],{},[598,4422,964],{},[762,4424,1165],{},[602,4426,609],{},[762,4428,3533],{},[602,4430,1039],{},[762,4432,4433],{},"1,8125",[602,4435,609],{},[762,4437,4438],{},"2,3125",[630,4440,4441],{"encoding":632},"k_1 = 1, \\quad k_2 = 0{,}25 + 1{,}25 = 1{,}5, \\quad k_3 = 0{,}25 + 1{,}375 = 1{,}625, \\quad k_4 = 0{,}5 + 1{,}8125 = 2{,}3125",[576,4443,4445,4500,4568,4596,4623,4699,4726,4754,4831,4858,4886],{"className":4444,"ariaHidden":622},[637],[576,4446,4448,4451,4491,4494,4497],{"className":4447},[641],[576,4449],{"className":4450,"style":2617},[645],[576,4452,4454,4457],{"className":4453},[650],[576,4455,964],{"className":4456,"style":1263},[650,654],[576,4458,4460],{"className":4459},[659],[576,4461,4463,4483],{"className":4462},[663,805],[576,4464,4466,4480],{"className":4465},[667],[576,4467,4469],{"className":4468,"style":812},[671],[576,4470,4471,4474],{"style":1278},[576,4472],{"className":4473,"style":680},[679],[576,4475,4477],{"className":4476},[684,685,686,687],[576,4478,967],{"className":4479},[650,687],[576,4481,829],{"className":4482},[828],[576,4484,4486],{"className":4485},[667],[576,4487,4489],{"className":4488,"style":836},[671],[576,4490],{},[576,4492],{"className":4493,"style":698},[697],[576,4495,609],{"className":4496},[702],[576,4498],{"className":4499,"style":698},[697],[576,4501,4503,4506,4509,4512,4516,4519,4559,4562,4565],{"className":4502},[641],[576,4504],{"className":4505,"style":2544},[645],[576,4507,967],{"className":4508},[650],[576,4510,623],{"className":4511},[727],[576,4513],{"className":4514,"style":4515},[697],"margin-right:1em;",[576,4517],{"className":4518,"style":731},[697],[576,4520,4522,4525],{"className":4521},[650],[576,4523,964],{"className":4524,"style":1263},[650,654],[576,4526,4528],{"className":4527},[659],[576,4529,4531,4551],{"className":4530},[663,805],[576,4532,4534,4548],{"className":4533},[667],[576,4535,4537],{"className":4536,"style":812},[671],[576,4538,4539,4542],{"style":1278},[576,4540],{"className":4541,"style":680},[679],[576,4543,4545],{"className":4544},[684,685,686,687],[576,4546,1015],{"className":4547},[650,687],[576,4549,829],{"className":4550},[828],[576,4552,4554],{"className":4553},[667],[576,4555,4557],{"className":4556,"style":836},[671],[576,4558],{},[576,4560],{"className":4561,"style":698},[697],[576,4563,609],{"className":4564},[702],[576,4566],{"className":4567,"style":698},[697],[576,4569,4571,4574,4577,4583,4587,4590,4593],{"className":4570},[641],[576,4572],{"className":4573,"style":2487},[645],[576,4575,764],{"className":4576},[650],[576,4578,4580],{"className":4579},[650],[576,4581,623],{"className":4582},[727],[576,4584,4586],{"className":4585},[650],"25",[576,4588],{"className":4589,"style":1664},[697],[576,4591,1039],{"className":4592},[1668],[576,4594],{"className":4595,"style":1664},[697],[576,4597,4599,4602,4605,4611,4614,4617,4620],{"className":4598},[641],[576,4600],{"className":4601,"style":2487},[645],[576,4603,967],{"className":4604},[650],[576,4606,4608],{"className":4607},[650],[576,4609,623],{"className":4610},[727],[576,4612,4586],{"className":4613},[650],[576,4615],{"className":4616,"style":698},[697],[576,4618,609],{"className":4619},[702],[576,4621],{"className":4622,"style":698},[697],[576,4624,4626,4629,4632,4638,4641,4644,4647,4650,4690,4693,4696],{"className":4625},[641],[576,4627],{"className":4628,"style":2544},[645],[576,4630,967],{"className":4631},[650],[576,4633,4635],{"className":4634},[650],[576,4636,623],{"className":4637},[727],[576,4639,3576],{"className":4640},[650],[576,4642,623],{"className":4643},[727],[576,4645],{"className":4646,"style":4515},[697],[576,4648],{"className":4649,"style":731},[697],[576,4651,4653,4656],{"className":4652},[650],[576,4654,964],{"className":4655,"style":1263},[650,654],[576,4657,4659],{"className":4658},[659],[576,4660,4662,4682],{"className":4661},[663,805],[576,4663,4665,4679],{"className":4664},[667],[576,4666,4668],{"className":4667,"style":812},[671],[576,4669,4670,4673],{"style":1278},[576,4671],{"className":4672,"style":680},[679],[576,4674,4676],{"className":4675},[684,685,686,687],[576,4677,1092],{"className":4678},[650,687],[576,4680,829],{"className":4681},[828],[576,4683,4685],{"className":4684},[667],[576,4686,4688],{"className":4687,"style":836},[671],[576,4689],{},[576,4691],{"className":4692,"style":698},[697],[576,4694,609],{"className":4695},[702],[576,4697],{"className":4698,"style":698},[697],[576,4700,4702,4705,4708,4714,4717,4720,4723],{"className":4701},[641],[576,4703],{"className":4704,"style":2487},[645],[576,4706,764],{"className":4707},[650],[576,4709,4711],{"className":4710},[650],[576,4712,623],{"className":4713},[727],[576,4715,4586],{"className":4716},[650],[576,4718],{"className":4719,"style":1664},[697],[576,4721,1039],{"className":4722},[1668],[576,4724],{"className":4725,"style":1664},[697],[576,4727,4729,4732,4735,4741,4745,4748,4751],{"className":4728},[641],[576,4730],{"className":4731,"style":2487},[645],[576,4733,967],{"className":4734},[650],[576,4736,4738],{"className":4737},[650],[576,4739,623],{"className":4740},[727],[576,4742,4744],{"className":4743},[650],"375",[576,4746],{"className":4747,"style":698},[697],[576,4749,609],{"className":4750},[702],[576,4752],{"className":4753,"style":698},[697],[576,4755,4757,4760,4763,4769,4773,4776,4779,4782,4822,4825,4828],{"className":4756},[641],[576,4758],{"className":4759,"style":2544},[645],[576,4761,967],{"className":4762},[650],[576,4764,4766],{"className":4765},[650],[576,4767,623],{"className":4768},[727],[576,4770,4772],{"className":4771},[650],"625",[576,4774,623],{"className":4775},[727],[576,4777],{"className":4778,"style":4515},[697],[576,4780],{"className":4781,"style":731},[697],[576,4783,4785,4788],{"className":4784},[650],[576,4786,964],{"className":4787,"style":1263},[650,654],[576,4789,4791],{"className":4790},[659],[576,4792,4794,4814],{"className":4793},[663,805],[576,4795,4797,4811],{"className":4796},[667],[576,4798,4800],{"className":4799,"style":812},[671],[576,4801,4802,4805],{"style":1278},[576,4803],{"className":4804,"style":680},[679],[576,4806,4808],{"className":4807},[684,685,686,687],[576,4809,1165],{"className":4810},[650,687],[576,4812,829],{"className":4813},[828],[576,4815,4817],{"className":4816},[667],[576,4818,4820],{"className":4819,"style":836},[671],[576,4821],{},[576,4823],{"className":4824,"style":698},[697],[576,4826,609],{"className":4827},[702],[576,4829],{"className":4830,"style":698},[697],[576,4832,4834,4837,4840,4846,4849,4852,4855],{"className":4833},[641],[576,4835],{"className":4836,"style":2487},[645],[576,4838,764],{"className":4839},[650],[576,4841,4843],{"className":4842},[650],[576,4844,623],{"className":4845},[727],[576,4847,3576],{"className":4848},[650],[576,4850],{"className":4851,"style":1664},[697],[576,4853,1039],{"className":4854},[1668],[576,4856],{"className":4857,"style":1664},[697],[576,4859,4861,4864,4867,4873,4877,4880,4883],{"className":4860},[641],[576,4862],{"className":4863,"style":2487},[645],[576,4865,967],{"className":4866},[650],[576,4868,4870],{"className":4869},[650],[576,4871,623],{"className":4872},[727],[576,4874,4876],{"className":4875},[650],"8125",[576,4878],{"className":4879,"style":698},[697],[576,4881,609],{"className":4882},[702],[576,4884],{"className":4885,"style":698},[697],[576,4887,4889,4892,4895,4901],{"className":4888},[641],[576,4890],{"className":4891,"style":2487},[645],[576,4893,1015],{"className":4894},[650],[576,4896,4898],{"className":4897},[650],[576,4899,623],{"className":4900},[727],[576,4902,4904],{"className":4903},[650],"3125",[576,4906,4908],{"className":4907},[934],[576,4909,4911,4967],{"className":4910},[579],[576,4912,4914],{"className":4913},[583],[585,4915,4916],{"xmlns":587,"display":943},[589,4917,4918,4964],{},[592,4919,4920,4926,4928,4930,4932,4938,4959,4961],{},[757,4921,4922,4924],{},[598,4923,600],{},[762,4925,967],{},[602,4927,609],{},[762,4929,967],{},[602,4931,1039],{},[1043,4933,4934,4936],{},[762,4935,3533],{},[762,4937,3132],{},[592,4939,4940,4942,4944,4946,4948,4950,4953,4955,4957],{},[602,4941,616],{"fence":622},[762,4943,967],{},[602,4945,1039],{},[762,4947,1092],{},[602,4949,1039],{},[762,4951,4952],{},"3,25",[602,4954,1039],{},[762,4956,4438],{},[602,4958,628],{"fence":622},[602,4960,609],{},[762,4962,4963],{},"1,796875",[630,4965,4966],{"encoding":632},"y_1 = 1 + \\frac{0{,}5}{6} \\left( 1 + 3 + 3{,}25 + 2{,}3125 \\right) = 1{,}796875",[576,4968,4970,5025,5043,5206],{"className":4969,"ariaHidden":622},[637],[576,4971,4973,4976,5016,5019,5022],{"className":4972},[641],[576,4974],{"className":4975,"style":857},[645],[576,4977,4979,4982],{"className":4978},[650],[576,4980,600],{"className":4981,"style":655},[650,654],[576,4983,4985],{"className":4984},[659],[576,4986,4988,5008],{"className":4987},[663,805],[576,4989,4991,5005],{"className":4990},[667],[576,4992,4994],{"className":4993,"style":812},[671],[576,4995,4996,4999],{"style":878},[576,4997],{"className":4998,"style":680},[679],[576,5000,5002],{"className":5001},[684,685,686,687],[576,5003,967],{"className":5004},[650,687],[576,5006,829],{"className":5007},[828],[576,5009,5011],{"className":5010},[667],[576,5012,5014],{"className":5013,"style":836},[671],[576,5015],{},[576,5017],{"className":5018,"style":698},[697],[576,5020,609],{"className":5021},[702],[576,5023],{"className":5024,"style":698},[697],[576,5026,5028,5031,5034,5037,5040],{"className":5027},[641],[576,5029],{"className":5030,"style":3158},[645],[576,5032,967],{"className":5033},[650],[576,5035],{"className":5036,"style":1664},[697],[576,5038,1039],{"className":5039},[1668],[576,5041],{"className":5042,"style":1664},[697],[576,5044,5046,5050,5125,5128,5197,5200,5203],{"className":5045},[641],[576,5047],{"className":5048,"style":5049},[645],"height:2.0074em;vertical-align:-0.686em;",[576,5051,5053,5056,5122],{"className":5052},[650],[576,5054],{"className":5055},[720,1678],[576,5057,5059],{"className":5058},[1043],[576,5060,5062,5113],{"className":5061},[663,805],[576,5063,5065,5110],{"className":5064},[667],[576,5066,5069,5081,5089],{"className":5067,"style":5068},[671],"height:1.3214em;",[576,5070,5072,5075],{"style":5071},"top:-2.314em;",[576,5073],{"className":5074,"style":1253},[679],[576,5076,5078],{"className":5077},[650],[576,5079,3132],{"className":5080},[650],[576,5082,5083,5086],{"style":1709},[576,5084],{"className":5085,"style":1253},[679],[576,5087],{"className":5088,"style":1717},[1716],[576,5090,5092,5095],{"style":5091},"top:-3.677em;",[576,5093],{"className":5094,"style":1253},[679],[576,5096,5098,5101,5107],{"className":5097},[650],[576,5099,764],{"className":5100},[650],[576,5102,5104],{"className":5103},[650],[576,5105,623],{"className":5106},[727],[576,5108,3576],{"className":5109},[650],[576,5111,829],{"className":5112},[828],[576,5114,5116],{"className":5115},[667],[576,5117,5120],{"className":5118,"style":5119},[671],"height:0.686em;",[576,5121],{},[576,5123],{"className":5124},[738,1678],[576,5126],{"className":5127,"style":731},[697],[576,5129,5132,5137,5140,5143,5146,5149,5152,5155,5158,5161,5164,5170,5173,5176,5179,5182,5185,5191,5194],{"className":5130},[5131],"minner",[576,5133,616],{"className":5134,"style":5136},[720,5135],"delimcenter","top:0em;",[576,5138,967],{"className":5139},[650],[576,5141],{"className":5142,"style":1664},[697],[576,5144,1039],{"className":5145},[1668],[576,5147],{"className":5148,"style":1664},[697],[576,5150,1092],{"className":5151},[650],[576,5153],{"className":5154,"style":1664},[697],[576,5156,1039],{"className":5157},[1668],[576,5159],{"className":5160,"style":1664},[697],[576,5162,1092],{"className":5163},[650],[576,5165,5167],{"className":5166},[650],[576,5168,623],{"className":5169},[727],[576,5171,4586],{"className":5172},[650],[576,5174],{"className":5175,"style":1664},[697],[576,5177,1039],{"className":5178},[1668],[576,5180],{"className":5181,"style":1664},[697],[576,5183,1015],{"className":5184},[650],[576,5186,5188],{"className":5187},[650],[576,5189,623],{"className":5190},[727],[576,5192,4904],{"className":5193},[650],[576,5195,628],{"className":5196,"style":5136},[738,5135],[576,5198],{"className":5199,"style":698},[697],[576,5201,609],{"className":5202},[702],[576,5204],{"className":5205,"style":698},[697],[576,5207,5209,5212,5215,5221],{"className":5208},[641],[576,5210],{"className":5211,"style":2487},[645],[576,5213,967],{"className":5214},[650],[576,5216,5218],{"className":5217},[650],[576,5219,623],{"className":5220},[727],[576,5222,5224],{"className":5223},[650],"796875",[564,5226,5227,5228,5421,5422,5462,5463,5491,5492,4104],{},"Exakt ist ",[576,5229,5231,5273],{"className":5230},[579],[576,5232,5234],{"className":5233},[583],[585,5235,5236],{"xmlns":587},[589,5237,5238,5270],{},[592,5239,5240,5242,5244,5246,5248,5250,5252,5254,5260,5263,5265,5267],{},[598,5241,600],{},[602,5243,616],{"stretchy":615},[762,5245,3533],{},[602,5247,628],{"stretchy":615},[602,5249,609],{},[762,5251,1015],{},[992,5253,1071],{},[595,5255,5256,5258],{},[598,5257,3044],{},[762,5259,3533],{},[602,5261,5262],{},"−",[762,5264,4390],{},[602,5266,3650],{},[762,5268,5269],{},"1,7974",[630,5271,5272],{"encoding":632},"y(0{,}5) = 2\\,e^{0{,}5} - 1{,}5 \\approx 1{,}7974",[576,5274,5276,5312,5375,5402],{"className":5275,"ariaHidden":622},[637],[576,5277,5279,5282,5285,5288,5291,5297,5300,5303,5306,5309],{"className":5278},[641],[576,5280],{"className":5281,"style":712},[645],[576,5283,600],{"className":5284,"style":655},[650,654],[576,5286,616],{"className":5287},[720],[576,5289,764],{"className":5290},[650],[576,5292,5294],{"className":5293},[650],[576,5295,623],{"className":5296},[727],[576,5298,3576],{"className":5299},[650],[576,5301,628],{"className":5302},[738],[576,5304],{"className":5305,"style":698},[697],[576,5307,609],{"className":5308},[702],[576,5310],{"className":5311,"style":698},[697],[576,5313,5315,5319,5322,5325,5366,5369,5372],{"className":5314},[641],[576,5316],{"className":5317,"style":5318},[645],"height:0.8974em;vertical-align:-0.0833em;",[576,5320,1015],{"className":5321},[650],[576,5323],{"className":5324,"style":731},[697],[576,5326,5328,5331],{"className":5327},[650],[576,5329,3044],{"className":5330},[650,654],[576,5332,5334],{"className":5333},[659],[576,5335,5337],{"className":5336},[663],[576,5338,5340],{"className":5339},[667],[576,5341,5343],{"className":5342,"style":4023},[671],[576,5344,5345,5348],{"style":675},[576,5346],{"className":5347,"style":680},[679],[576,5349,5351],{"className":5350},[684,685,686,687],[576,5352,5354,5357,5363],{"className":5353},[650,687],[576,5355,764],{"className":5356},[650,687],[576,5358,5360],{"className":5359},[650,687],[576,5361,623],{"className":5362},[727,687],[576,5364,3576],{"className":5365},[650,687],[576,5367],{"className":5368,"style":1664},[697],[576,5370,5262],{"className":5371},[1668],[576,5373],{"className":5374,"style":1664},[697],[576,5376,5378,5381,5384,5390,5393,5396,5399],{"className":5377},[641],[576,5379],{"className":5380,"style":2487},[645],[576,5382,967],{"className":5383},[650],[576,5385,5387],{"className":5386},[650],[576,5388,623],{"className":5389},[727],[576,5391,3576],{"className":5392},[650],[576,5394],{"className":5395,"style":698},[697],[576,5397,3650],{"className":5398},[702],[576,5400],{"className":5401,"style":698},[697],[576,5403,5405,5408,5411,5417],{"className":5404},[641],[576,5406],{"className":5407,"style":2487},[645],[576,5409,967],{"className":5410},[650],[576,5412,5414],{"className":5413},[650],[576,5415,623],{"className":5416},[727],[576,5418,5420],{"className":5419},[650],"7974",". Der Fehler beträgt etwa ",[576,5423,5425,5440],{"className":5424},[579],[576,5426,5428],{"className":5427},[583],[585,5429,5430],{"xmlns":587},[589,5431,5432,5437],{},[592,5433,5434],{},[762,5435,5436],{},"0,0006",[630,5438,5439],{"encoding":632},"0{,}0006",[576,5441,5443],{"className":5442,"ariaHidden":622},[637],[576,5444,5446,5449,5452,5458],{"className":5445},[641],[576,5447],{"className":5448,"style":2487},[645],[576,5450,764],{"className":5451},[650],[576,5453,5455],{"className":5454},[650],[576,5456,623],{"className":5457},[727],[576,5459,5461],{"className":5460},[650],"0006",", beim Euler-Verfahren mit gleichem 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zählen doppelt, nicht die äußeren.",[2721,5682,5683,5684,5753,5754,5823],{},"In 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und 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den vollen statt den halben Schritt 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wieder 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statt 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einsetzen.",[2721,6038,6039,6040,6068,6069,6097,6098,6193,6194,4104],{},"Durch ",[576,6041,6043,6056],{"className":6042},[579],[576,6044,6046],{"className":6045},[583],[585,6047,6048],{"xmlns":587},[589,6049,6050,6054],{},[592,6051,6052],{},[762,6053,1165],{},[630,6055,1165],{"encoding":632},[576,6057,6059],{"className":6058,"ariaHidden":622},[637],[576,6060,6062,6065],{"className":6061},[641],[576,6063],{"className":6064,"style":3023},[645],[576,6066,1165],{"className":6067},[650]," statt durch ",[576,6070,6072,6085],{"className":6071},[579],[576,6073,6075],{"className":6074},[583],[585,6076,6077],{"xmlns":587},[589,6078,6079,6083],{},[592,6080,6081],{},[762,6082,3132],{},[630,6084,3132],{"encoding":632},[576,6086,6088],{"className":6087,"ariaHidden":622},[637],[576,6089,6091,6094],{"className":6090},[641],[576,6092],{"className":6093,"style":3023},[645],[576,6095,3132],{"className":6096},[650]," teilen. Die Gewichte ",[576,6099,6101,6127],{"className":6100},[579],[576,6102,6104],{"className":6103},[583],[585,6105,6106],{"xmlns":587},[589,6107,6108,6124],{},[592,6109,6110,6112,6114,6116,6118,6120,6122],{},[762,6111,967],{},[602,6113,1039],{},[762,6115,1015],{},[602,6117,1039],{},[762,6119,1015],{},[602,6121,1039],{},[762,6123,967],{},[630,6125,6126],{"encoding":632},"1 + 2 + 2 + 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ergeben ",[576,6195,6197,6210],{"className":6196},[579],[576,6198,6200],{"className":6199},[583],[585,6201,6202],{"xmlns":587},[589,6203,6204,6208],{},[592,6205,6206],{},[762,6207,3132],{},[630,6209,3132],{"encoding":632},[576,6211,6213],{"className":6212,"ariaHidden":622},[637],[576,6214,6216,6219],{"className":6215},[641],[576,6217],{"className":6218,"style":3023},[645],[576,6220,3132],{"className":6221},[650],{"title":6223,"searchDepth":6224,"depth":6224,"links":6225},"",2,[6226,6227,6228],{"id":561,"depth":6224,"text":562},{"id":2872,"depth":6224,"text":2873},{"id":5533,"depth":6224,"text":5534},[6230,6233,6236,6239,6242,6245,6248,6250,6253,6256,6259,6262,6265],{"lhs":6231,"rhs":6232},"y + h\u002F6*(y + 2*y*(1 + h\u002F2) + 2*y*(1 + h\u002F2*(1 + h\u002F2)) + y*(1 + h*(1 + h\u002F2*(1 + h\u002F2))))","y*(1 + h + h^2\u002F2 + h^3\u002F6 + h^4\u002F24)",{"lhs":6234,"rhs":6235},"1 + 0.5 + 0.5^2\u002F2 + 0.5^3\u002F6 + 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style=\"height:0.3011em;\">\u003Cspan style=\"top:-2.55em;margin-left:-0.0315em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"katex-sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">4\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mclose delimcenter\" style=\"top:0em;\">)\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>","university",{},true,"\u002Fde\u002Fformulas\u002Fmath\u002Fnumerics\u002Frunge-kutta-verfahren",[],[6281,6282,6283,6284],"euler-method","implicit-euler","simpson-rule","order-of-convergence",{"title":548,"description":6223},"de\u002Fformulas\u002Fmath\u002Fnumerics\u002Frunge-kutta-verfahren","Das klassische Runge-Kutta-Verfahren mittelt vier Steigungen pro Schritt und erreicht so die Konvergenzordnung 4.",[6289,6290,6291,6292],"differentialgleichung","anfangswertproblem","einschrittverfahren","runge-kutta",[6294,6298,6301,6304,6307,6310],{"symbol":6295,"meaning":6296,"symbolHtml":6297},"y_n","Näherung für die Lösung y(t_n)","\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmsub>\u003Cmi>y\u003C\u002Fmi>\u003Cmi>n\u003C\u002Fmi>\u003C\u002Fmsub>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">y_n\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.625em;vertical-align:-0.1944em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0359em;\">y\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.1514em;\">\u003Cspan style=\"top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"katex-sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mathnormal mtight\">n\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",{"symbol":914,"meaning":6299,"symbolHtml":6300},"Schrittweite","\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>h\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">h\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.6944em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">h\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",{"symbol":5916,"meaning":6302,"symbolHtml":6303},"Steigung am Anfang, k₁ = f(t_n, y_n)","\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmsub>\u003Cmi>k\u003C\u002Fmi>\u003Cmn>1\u003C\u002Fmn>\u003C\u002Fmsub>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">k_1\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.8444em;vertical-align:-0.15em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0315em;\">k\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3011em;\">\u003Cspan style=\"top:-2.55em;margin-left:-0.0315em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"katex-sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">1\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",{"symbol":5560,"meaning":6305,"symbolHtml":6306},"Steigung in der Mitte, k₂ = f(t_n + h\u002F2, y_n + h\u002F2 · k₁)","\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmsub>\u003Cmi>k\u003C\u002Fmi>\u003Cmn>2\u003C\u002Fmn>\u003C\u002Fmsub>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">k_2\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.8444em;vertical-align:-0.15em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0315em;\">k\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3011em;\">\u003Cspan style=\"top:-2.55em;margin-left:-0.0315em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"katex-sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">2\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",{"symbol":5630,"meaning":6308,"symbolHtml":6309},"verbesserte Steigung in der Mitte, k₃ = f(t_n + h\u002F2, y_n + h\u002F2 · k₂)","\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmsub>\u003Cmi>k\u003C\u002Fmi>\u003Cmn>3\u003C\u002Fmn>\u003C\u002Fmsub>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">k_3\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.8444em;vertical-align:-0.15em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0315em;\">k\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3011em;\">\u003Cspan style=\"top:-2.55em;margin-left:-0.0315em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"katex-sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">3\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",{"symbol":6311,"meaning":6312,"symbolHtml":6313},"k_4","Steigung am Ende, k₄ = f(t_n + h, y_n + h · k₃)","\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmsub>\u003Cmi>k\u003C\u002Fmi>\u003Cmn>4\u003C\u002Fmn>\u003C\u002Fmsub>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">k_4\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.8444em;vertical-align:-0.15em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0315em;\">k\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3011em;\">\u003Cspan style=\"top:-2.55em;margin-left:-0.0315em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"katex-sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">4\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>","DQx0mfXwNyuOax268XVKGp8HyAw0O_CBY7M7OJa9YNg",[6316,6319,6322,6325],{"stem":6317,"title":6318,"category":391},"de\u002Fformulas\u002Fmath\u002Fnumerics\u002Feuler-verfahren","Explizites Euler-Verfahren",{"stem":6320,"title":6321,"category":391},"de\u002Fformulas\u002Fmath\u002Fnumerics\u002Fimplizites-euler-verfahren","Implizites Euler-Verfahren",{"stem":6323,"title":6324,"category":391},"de\u002Fformulas\u002Fmath\u002Fnumerics\u002Fkonvergenzordnung","Konvergenzordnung",{"stem":6326,"title":6327,"category":391},"de\u002Fformulas\u002Fmath\u002Fnumerics\u002Fsimpsonregel","Simpsonregel",1793824265790]