2020-04-06T07:53:05-04:00
Using the classical fourth order Runge-Kutta method, find the approximate value of y(0.6) for the initial value problem dy/dx = sin xy ; y(0) =1 with the step size h =0.2
1
2020-04-13T12:50:59-0400
y n + 1 = y n + h ( k 1 6 + k 2 3 + k 3 3 + k 4 6 ) y_{n+1}=y_n+h(\frac {k_1}{6}+\frac{k_2}{3}+\frac{k_3}{3}+\frac {k_4}{6}) y n + 1 = y n + h ( 6 k 1 + 3 k 2 + 3 k 3 + 6 k 4 )
x n + 1 = x n + h x_{n+1}=x_n+h x n + 1 = x n + h
k 1 = f ( x n ; y n ) k_1=f(x_n;y_n) k 1 = f ( x n ; y n )
k 2 = f ( x n + h 2 ; y n + h k 1 2 ) k_2=f(x_n+\frac{h}{2}; y_n+\frac {hk_1}{2}) k 2 = f ( x n + 2 h ; y n + 2 h k 1 )
k 3 = f ( x n + h 3 ; y n + h k 2 2 ) k_3=f(x_n+\frac{h}{3}; y_n+\frac {hk_2}{2}) k 3 = f ( x n + 3 h ; y n + 2 h k 2 )
k 4 = f ( x n + h ; y n + h k 3 ) k_4=f(x_n+h; y_n+hk_3) k 4 = f ( x n + h ; y n + h k 3 )
Stability function:
z 4 24 + z 3 6 + z 2 2 + z + 1 \frac{z^4}{24}+\frac{z^3}{6}+\frac{z^2}{2}+z+1 24 z 4 + 6 z 3 + 2 z 2 + z + 1
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