81.8.6 problem 13

Internal problem ID [18716]
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 : 13
Date solved : Tuesday, January 28, 2025 at 12:12:15 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d x}y \left (x \right )+\frac {d}{d x}z \left (x \right )+6 y \left (x \right )&=0\\ \frac {d}{d x}z \left (x \right )+5 y \left (x \right )+z \left (x \right )&=0 \end{align*}

✓ Solution by Maple

Time used: 0.035 (sec). Leaf size: 55

dsolve([diff(y(x),x)+diff(z(x),x)+6*y(x)=0,diff(z(x),x)+5*y(x)+z(x)=0],singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.012 (sec). Leaf size: 81

DSolve[{D[y[x],x]+D[z[x],x]+6*y[x]==0,D[z[x],x]+5*y[x]+z[x]==0},{y[x],z[x]},x,IncludeSingularSolutions -> True]
 

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