81.8.2 problem 9

Internal problem ID [18712]
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 : 9
Date solved : Tuesday, January 28, 2025 at 12:12:12 PM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.020 (sec). Leaf size: 34

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

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

✓ Solution by Mathematica

Time used: 0.009 (sec). Leaf size: 145

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

\begin{align*} y(x)\to \frac {1}{146} e^{-\frac {1}{2} \left (7+\sqrt {73}\right ) x} \left (c_1 \left (\left (73+\sqrt {73}\right ) e^{\sqrt {73} x}+73-\sqrt {73}\right )+4 \sqrt {73} c_2 \left (e^{\sqrt {73} x}-1\right )\right ) \\ z(x)\to \frac {1}{146} e^{-\frac {1}{2} \left (7+\sqrt {73}\right ) x} \left (18 \sqrt {73} c_1 \left (e^{\sqrt {73} x}-1\right )-c_2 \left (\left (\sqrt {73}-73\right ) e^{\sqrt {73} x}-73-\sqrt {73}\right )\right ) \\ \end{align*}