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

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 Problem: Runge-Kutta 4 method y'=-2x-y, y(0)=1, xn=0.5, h=0.1 [ Calculator, Method and examples ]Solution:Your problem -> Runge-Kutta 4 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)=?Forth 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)=-0.105k_3=hf(x_0+h/2,y_0+k_2/2)=(0.1)f(0.05,0.9475)=(0.1)*(-1.0475)=-0.1048k_4=hf(x_0+h,y_0+k_3)=(0.1)f(0.1,0.8952)=(0.1)*(-1.0952)=-0.1095y_1=y_0+1/6(k_1+2k_2+2k_3+k_4)y_1=1+1/6[-0.1+2(-0.105)+2(-0.1048)+(-0.1095)]y_1=0.8952:.y(0.1)=0.8952Again taking (x_1,y_1) in place of (x_0,y_0) repeat the processk_1=hf(x_1,y_1)=(0.1)f(0.1,0.8952)=(0.1)*(-1.0952)=-0.1095k_2=hf(x_1+h/2,y_1+k_1/2)=(0.1)f(0.15,0.8404)=(0.1)*(-1.1404)=-0.114k_3=hf(x_1+h/2,y_1+k_2/2)=(0.1)f(0.15,0.8381)=(0.1)*(-1.1381)=-0.1138k_4=hf(x_1+h,y_1+k_3)=(0.1)f(0.2,0.7813)=(0.1)*(-1.1813)=-0.1181y_2=y_1+1/6(k_1+2k_2+2k_3+k_4)y_2=0.8952+1/6[-0.1095+2(-0.114)+2(-0.1138)+(-0.1181)]y_2=0.7813:.y(0.2)=0.7813Again taking (x_2,y_2) in place of (x_0,y_0) repeat the processk_1=hf(x_2,y_2)=(0.1)f(0.2,0.7813)=(0.1)*(-1.1813)=-0.1181k_2=hf(x_2+h/2,y_2+k_1/2)=(0.1)f(0.25,0.7222)=(0.1)*(-1.2222)=-0.1222k_3=hf(x_2+h/2,y_2+k_2/2)=(0.1)f(0.25,0.7202)=(0.1)*(-1.2202)=-0.122k_4=hf(x_2+h,y_2+k_3)=(0.1)f(0.3,0.6593)=(0.1)*(-1.2593)=-0.1259

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