81.8.7 problem 14

Internal problem ID [18717]
Book : A short course on differential equations. By Donald Francis Campbell. Maxmillan company. London. 1907
Section : Chapter VII. Ordinary differential equations in two dependent variables. Exercises at page 86
Problem number : 14
Date solved : Tuesday, January 28, 2025 at 12:12:16 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d x}z \left (x \right )+\frac {d}{d x}y \left (x \right )+5 y \left (x \right )-3 z \left (x \right )&=x +{\mathrm e}^{x}\\ \frac {d}{d x}y \left (x \right )+2 y \left (x \right )-z \left (x \right )&={\mathrm e}^{x} \end{align*}

✓ Solution by Maple

Time used: 0.035 (sec). Leaf size: 55

dsolve([diff(z(x),x)+diff(y(x),x)+5*y(x)-3*z(x)=x+exp(x),diff(y(x),x)+2*y(x)-z(x)=exp(x)],singsol=all)
 

\begin{align*} y \left (x \right ) &= \frac {{\mathrm e}^{x} c_{1}}{3}-\frac {{\mathrm e}^{x} x}{2}+c_{2} {\mathrm e}^{-x}+\frac {3 \,{\mathrm e}^{x}}{4}-x \\ z \left (x \right ) &= c_{2} {\mathrm e}^{-x}+{\mathrm e}^{x} c_{1} -\frac {3 \,{\mathrm e}^{x} x}{2}+\frac {3 \,{\mathrm e}^{x}}{4}-2 x -1 \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.209 (sec). Leaf size: 98

DSolve[{D[z[x],x]+D[y[x],x]+5*y[x]-3*z[x]==x+Exp[x],D[y[x],x]+2*y[x]-z[x]==Exp[x]},{y[x],z[x]},x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \frac {1}{4} e^{-x} \left (-4 e^x x+e^{2 x} (-2 x+3-2 c_1+2 c_2)+6 c_1-2 c_2\right ) \\ z(x)\to \frac {1}{4} e^{-x} \left (-4 e^x (2 x+1)+e^{2 x} (-6 x+3-6 c_1+6 c_2)+6 c_1-2 c_2\right ) \\ \end{align*}