81.8.3 problem 10

Internal problem ID [18713]
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 : 10
Date solved : Tuesday, January 28, 2025 at 12:12:12 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d x}y \left (x \right )+3 y \left (x \right )+z \left (x \right )&=0\\ \frac {d}{d x}z \left (x \right )+3 y \left (x \right )+5 z \left (x \right )&=0 \end{align*}

✓ Solution by Maple

Time used: 0.017 (sec). Leaf size: 35

dsolve([diff(y(x),x)+3*y(x)+z(x)=0,diff(z(x),x)+3*y(x)+5*z(x)=0],singsol=all)
 

\begin{align*} y \left (x \right ) &= {\mathrm e}^{-2 x} c_{1} +c_{2} {\mathrm e}^{-6 x} \\ z \left (x \right ) &= -{\mathrm e}^{-2 x} c_{1} +3 c_{2} {\mathrm e}^{-6 x} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.003 (sec). Leaf size: 71

DSolve[{D[y[x],x]+3*y[x]+z[x]==0,D[z[x],x]+3*y[x]+5*z[x]==0},{y[x],z[x]},x,IncludeSingularSolutions -> True]
 

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