81.8.5 problem 12

Internal problem ID [18715]
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 : 12
Date solved : Tuesday, January 28, 2025 at 12:12:14 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d x}y \left (x \right )-3 y \left (x \right )-2 z \left (x \right )&=0\\ \frac {d}{d x}z \left (x \right )+y \left (x \right )-2 z \left (x \right )&=0 \end{align*}

✓ Solution by Maple

Time used: 0.021 (sec). Leaf size: 81

dsolve([diff(y(x),x)-3*y(x)-2*z(x)=0,diff(z(x),x)+y(x)-2*z(x)=0],singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.016 (sec). Leaf size: 111

DSolve[{D[y[x],x]-3*y[x]-2*z[x]==0,D[z[x],x]+y[x]-2*z[x]==0},{y[x],z[x]},x,IncludeSingularSolutions -> True]
 

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