91Ó°ÊÓ

Find \(y(0.4)\) if \(y^{\prime}=(x+y)^{2}\) and \(y(0)=1\) using the Runge-Kutta method of order 4 . Take (a) \(h=0.2\) and (b) \(h=0.1\)

Express the following equations as a set of first-order equations: (a) \(\frac{d^{2} y}{d x^{2}}+2 \frac{d y}{d x}+3 y=0\) (b) \(\frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}+8 \frac{\mathrm{d} y}{\mathrm{~d} x}+9 y=0\) (c) \(\frac{\mathrm{d}^{2} x}{\mathrm{~d} t^{2}}+4 \frac{\mathrm{d} x}{\mathrm{~d} t}+6 \mathrm{x}=0\) (d) \(\frac{\mathrm{d}^{2} y}{d t^{2}}+6 \frac{\mathrm{d} y}{\mathrm{~d} t}+7 y=0\) (e) \(\frac{\mathrm{d}^{3} y}{\mathrm{~d} x^{3}}+6 \frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}+2 \frac{\mathrm{d} y}{\mathrm{~d} x}+\mathrm{y}=0\) (f) \(\frac{\mathrm{d}^{3} x}{\mathrm{~d} t^{3}}+2 \frac{\mathrm{d}^{2} x}{\mathrm{~d} t^{2}}+4 \frac{\mathrm{d} x}{\mathrm{~d} t}+2 x=0\)

Express the following coupled first-order equations as a single second-order differential cquation:(a) \(\frac{\mathrm{dy}_{1}}{\mathrm{dx}}=y_{1}+y_{2}, \frac{\mathrm{d} y_{2}}{\mathrm{~d} x}=2 \mathrm{y}_{1}-2 y_{2}\) (b) \(\frac{\mathrm{dy}_{1}}{\mathrm{~d} x}=2 y_{1}-y_{2}, \frac{\mathrm{d} y_{2}}{\mathrm{~d} x}=4 y_{1}+y_{2}\) (c) \(\frac{\mathrm{d} x_{1}}{\mathrm{~d} t}=3 x_{1}+x_{2}, \frac{\mathrm{d} x_{2}}{\mathrm{~d} t}=2 x_{1}-3 x_{2}\) (d) \(\frac{\mathrm{d} y_{1}}{\mathrm{~d} t}=2 \mathrm{y}_{1}+4 \mathrm{y}_{2}, \frac{\mathrm{dy}_{2}}{\mathrm{~d} t}=6 \mathrm{y}_{1}-7 \mathrm{y}_{2}\)

Find \(y(0.5)\) if \(y^{\prime}=x+y, y(0)=0 .\) Use \(h=0.25\) and \(h=0.1 .\) Find the true solution for comparison.