81.8.4 problem 11

Internal problem ID [18714]
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 : 11
Date solved : Tuesday, January 28, 2025 at 12:12:13 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 )+2 y \left (x \right )-4 z \left (x \right )&=0 \end{align*}

✓ Solution by Maple

Time used: 0.023 (sec). Leaf size: 85

dsolve([diff(y(x),x)+3*y(x)+2*z(x)=0,diff(z(x),x)+2*y(x)-4*z(x)=0],singsol=all)
 

\begin{align*} y \left (x \right ) &= c_{1} {\mathrm e}^{\frac {\left (1+\sqrt {65}\right ) x}{2}}+c_{2} {\mathrm e}^{-\frac {\left (-1+\sqrt {65}\right ) x}{2}} \\ z \left (x \right ) &= -\frac {c_{1} {\mathrm e}^{\frac {\left (1+\sqrt {65}\right ) x}{2}} \sqrt {65}}{4}+\frac {c_{2} {\mathrm e}^{-\frac {\left (-1+\sqrt {65}\right ) x}{2}} \sqrt {65}}{4}-\frac {7 c_{1} {\mathrm e}^{\frac {\left (1+\sqrt {65}\right ) x}{2}}}{4}-\frac {7 c_{2} {\mathrm e}^{-\frac {\left (-1+\sqrt {65}\right ) x}{2}}}{4} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.011 (sec). Leaf size: 152

DSolve[{D[y[x],x]+3*y[x]+2*z[x]==0,D[z[x],x]+2*y[x]-4*z[x]==0},{y[x],z[x]},x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to -\frac {1}{130} e^{\frac {1}{2} \left (x-\sqrt {65} x\right )} \left (c_1 \left (\left (7 \sqrt {65}-65\right ) e^{\sqrt {65} x}-65-7 \sqrt {65}\right )+4 \sqrt {65} c_2 \left (e^{\sqrt {65} x}-1\right )\right ) \\ z(x)\to \frac {1}{130} e^{\frac {1}{2} \left (x-\sqrt {65} x\right )} \left (c_2 \left (\left (65+7 \sqrt {65}\right ) e^{\sqrt {65} x}+65-7 \sqrt {65}\right )-4 \sqrt {65} c_1 \left (e^{\sqrt {65} x}-1\right )\right ) \\ \end{align*}