81.8.12 problem 19

Internal problem ID [18722]
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 : 19
Date solved : Tuesday, January 28, 2025 at 12:12:21 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d x}y \left (x \right )-2 y \left (x \right )-2 z \left (x \right )&={\mathrm e}^{3 x}\\ \frac {d}{d x}z \left (x \right )+5 y \left (x \right )-2 z \left (x \right )&={\mathrm e}^{4 x} \end{align*}

✓ Solution by Maple

Time used: 0.090 (sec). Leaf size: 89

dsolve([diff(y(x),x)-2*y(x)-2*z(x)=exp(3*x),diff(z(x),x)+5*y(x)-2*z(x)=exp(4*x)],singsol=all)
 

\begin{align*} y \left (x \right ) &= {\mathrm e}^{2 x} \sin \left (\sqrt {10}\, x \right ) c_{2} +{\mathrm e}^{2 x} \cos \left (\sqrt {10}\, x \right ) c_{1} +\frac {{\mathrm e}^{4 x}}{7}+\frac {{\mathrm e}^{3 x}}{11} \\ z \left (x \right ) &= -\frac {5 \,{\mathrm e}^{3 x}}{11}+\frac {{\mathrm e}^{2 x} \sqrt {10}\, \cos \left (\sqrt {10}\, x \right ) c_{2}}{2}-\frac {{\mathrm e}^{2 x} \sqrt {10}\, \sin \left (\sqrt {10}\, x \right ) c_{1}}{2}+\frac {{\mathrm e}^{4 x}}{7} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.248 (sec). Leaf size: 110

DSolve[{D[y[x],x]-2*y[x]-2*z[x]==Exp[3*x],D[z[x],x]+5*y[x]-2*z[x]==Exp[4*x]},{y[x],z[x]},x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \frac {1}{385} e^{2 x} \left (5 e^x \left (11 e^x+7\right )+385 c_1 \cos \left (\sqrt {10} x\right )+77 \sqrt {10} c_2 \sin \left (\sqrt {10} x\right )\right ) \\ z(x)\to \frac {1}{154} e^{2 x} \left (2 e^x \left (11 e^x-35\right )+154 c_2 \cos \left (\sqrt {10} x\right )-77 \sqrt {10} c_1 \sin \left (\sqrt {10} x\right )\right ) \\ \end{align*}