81.8.11 problem 18

Internal problem ID [18721]
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 : 18
Date solved : Tuesday, January 28, 2025 at 12:12:20 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d x}z \left (x \right )+7 y \left (x \right )-9 z \left (x \right )&={\mathrm e}^{x}\\ \frac {d}{d x}y \left (x \right )-y \left (x \right )-3 z \left (x \right )&={\mathrm e}^{2 x} \end{align*}

✓ Solution by Maple

Time used: 0.092 (sec). Leaf size: 107

dsolve([diff(z(x),x)+7*y(x)-9*z(x)=exp(x),diff(y(x),x)-y(x)-3*z(x)=exp(2*x)],singsol=all)
 

\begin{align*} y \left (x \right ) &= {\mathrm e}^{5 x} \sin \left (\sqrt {5}\, x \right ) c_{2} +{\mathrm e}^{5 x} \cos \left (\sqrt {5}\, x \right ) c_{1} -\frac {{\mathrm e}^{2 x}}{2}+\frac {{\mathrm e}^{x}}{7} \\ z \left (x \right ) &= -\frac {{\mathrm e}^{2 x}}{2}+\frac {4 \,{\mathrm e}^{5 x} \sin \left (\sqrt {5}\, x \right ) c_{2}}{3}+\frac {{\mathrm e}^{5 x} \sqrt {5}\, \cos \left (\sqrt {5}\, x \right ) c_{2}}{3}+\frac {4 \,{\mathrm e}^{5 x} \cos \left (\sqrt {5}\, x \right ) c_{1}}{3}-\frac {{\mathrm e}^{5 x} \sqrt {5}\, \sin \left (\sqrt {5}\, x \right ) c_{1}}{3} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.653 (sec). Leaf size: 123

DSolve[{D[z[x],x]+7*y[x]-9*z[x]==Exp[x],D[y[x],x]-y[x]-3*z[x]==Exp[2*x]},{y[x],z[x]},x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \frac {1}{14} e^x \left (2-7 e^x\right )+c_1 e^{5 x} \cos \left (\sqrt {5} x\right )-\frac {(4 c_1-3 c_2) e^{5 x} \sin \left (\sqrt {5} x\right )}{\sqrt {5}} \\ z(x)\to -\frac {e^{2 x}}{2}+c_2 e^{5 x} \cos \left (\sqrt {5} x\right )-\frac {(7 c_1-4 c_2) e^{5 x} \sin \left (\sqrt {5} x\right )}{\sqrt {5}} \\ \end{align*}