Home > Numerical methods calculators > Numerical differential equation using Euler, Runge-kutta (Rk2, Rk3, Rk4) methods calculator

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 Problem: Runge-Kutta 2 method y'=-2x-y, y(0)=1, xn=0.5, h=0.1 [ Calculator, Method and examples ]Solution:Your problem -> Runge-Kutta 2 method y'=-2x-y, y(0)=1, xn=0.5, h=0.1Given y'=-2x-y, y(0)=1, h=0.1, y(0.5)=?Second order R-K methodk_1=hf(x_0,y_0)=(0.1)f(0,1)=(0.1)*(-1)=-0.1k_2=hf(x_0+h/2,y_0+k_1/2)=(0.1)f(0.05,0.95)=(0.1)*(-1.05)=-0.105y_1=y_0+k_2=1-0.105=0.895:.y(0.1)=0.895Again taking (x_1,y_1) in place of (x_0,y_0) repeat the processk_1=hf(x_1,y_1)=(0.1)f(0.1,0.895)=(0.1)*(-1.095)=-0.1095k_2=hf(x_1+h/2,y_1+k_1/2)=(0.1)f(0.15,0.8402)=(0.1)*(-1.1402)=-0.114y_2=y_1+k_2=0.895-0.114=0.781:.y(0.2)=0.781Again taking (x_2,y_2) in place of (x_0,y_0) repeat the processk_1=hf(x_2,y_2)=(0.1)f(0.2,0.781)=(0.1)*(-1.181)=-0.1181k_2=hf(x_2+h/2,y_2+k_1/2)=(0.1)f(0.25,0.7219)=(0.1)*(-1.2219)=-0.1222

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