81.8.8 problem 15

Internal problem ID [18718]
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 : 15
Date solved : Tuesday, January 28, 2025 at 12:12:17 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d x}z \left (x \right )+y \left (x \right )+3 z \left (x \right )&={\mathrm e}^{x}\\ \frac {d}{d x}y \left (x \right )+3 y \left (x \right )+4 z \left (x \right )&={\mathrm e}^{2 x} \end{align*}

✓ Solution by Maple

Time used: 0.039 (sec). Leaf size: 55

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

\begin{align*} y \left (x \right ) &= {\mathrm e}^{-5 x} c_{2} +{\mathrm e}^{-x} c_{1} -\frac {{\mathrm e}^{x}}{3}+\frac {5 \,{\mathrm e}^{2 x}}{21} \\ z \left (x \right ) &= -\frac {{\mathrm e}^{2 x}}{21}+\frac {{\mathrm e}^{-5 x} c_{2}}{2}-\frac {{\mathrm e}^{-x} c_{1}}{2}+\frac {{\mathrm e}^{x}}{3} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.065 (sec). Leaf size: 103

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

\begin{align*} y(x)\to -\frac {e^x}{3}+\frac {5 e^{2 x}}{21}+\frac {1}{2} (c_1-2 c_2) e^{-x}+\left (\frac {c_1}{2}+c_2\right ) e^{-5 x} \\ z(x)\to \frac {1}{84} e^{-5 x} \left (28 e^{6 x}-4 e^{7 x}-21 (c_1-2 c_2) e^{4 x}+21 (c_1+2 c_2)\right ) \\ \end{align*}