Use the RungeKutta Method of order 4 with h 025 to approxim

Use the Runge-Kutta Method of order 4 with h = 0.25 to approximate (to 6 decimal places) the solution of y\' = -2xy, y(0) = 3 on 0 lessthanorequalto x lessthanorequalto 1. Compare your approximations with the true values by calculating the errors at each step.

Solution

y\' = -2*x*y = f(x,y)

y(0) =y_n = 3, x_n = 0, h = 0.25

K₁ = h*(x_n ,y_n ) = 0.25*(-2*0*3) = 0

K₂ = h*(x_n+(h/2) ,y_n + (K₁/2)) = 0.25*(-2*0.125*3) = -0.1875

K₃ = h*(x_n+(h/2) ,y_n + (K₂/2)) = 0.25*(-2*0.125*2.90625) = -0.18164

K₄ = h*(x_n+h ,y_n + K₁) = 0.25*(-2*0.25*2.81836) = -0.352295

y\' = y(0) + [(K₁ +2*K₂ +2*K₃ +K₄ )*h]/6 = 2.9545593