81.8.10 problem 17

Internal problem ID [18720]
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 : 17
Date solved : Tuesday, January 28, 2025 at 12:12:19 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d x}z \left (x \right )+5 y \left (x \right )-2 z \left (x \right )&=x\\ \frac {d}{d x}y \left (x \right )+4 y \left (x \right )+z \left (x \right )&=x \end{align*}

✓ Solution by Maple

Time used: 0.020 (sec). Leaf size: 89

dsolve([diff(z(x),x)+5*y(x)-2*z(x)=x,diff(y(x),x)+4*y(x)+z(x)=x],singsol=all)
 

\begin{align*} y \left (x \right ) &= {\mathrm e}^{\left (-1+\sqrt {14}\right ) x} c_{2} +{\mathrm e}^{-\left (1+\sqrt {14}\right ) x} c_{1} +\frac {3 x}{13}-\frac {7}{169} \\ z \left (x \right ) &= -{\mathrm e}^{\left (-1+\sqrt {14}\right ) x} c_{2} \sqrt {14}+{\mathrm e}^{-\left (1+\sqrt {14}\right ) x} c_{1} \sqrt {14}-3 \,{\mathrm e}^{\left (-1+\sqrt {14}\right ) x} c_{2} -3 \,{\mathrm e}^{-\left (1+\sqrt {14}\right ) x} c_{1} +\frac {x}{13}-\frac {11}{169} \\ \end{align*}

✓ Solution by Mathematica

Time used: 1.464 (sec). Leaf size: 174

DSolve[{D[z[x],x]+5*y[x]-2*z[x]==x,D[y[x],x]+4*y[x]+z[x]==x},{y[x],z[x]},x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \frac {3 x}{13}-\frac {1}{28} \left (\left (3 \sqrt {14}-14\right ) c_1+\sqrt {14} c_2\right ) e^{\left (\sqrt {14}-1\right ) x}+\frac {1}{28} \left (\left (14+3 \sqrt {14}\right ) c_1+\sqrt {14} c_2\right ) e^{-\left (\left (1+\sqrt {14}\right ) x\right )}-\frac {7}{169} \\ z(x)\to \frac {x}{13}+\frac {1}{28} \left (5 \sqrt {14} c_1+\left (14-3 \sqrt {14}\right ) c_2\right ) e^{-\left (\left (1+\sqrt {14}\right ) x\right )}+\frac {1}{28} \left (\left (14+3 \sqrt {14}\right ) c_2-5 \sqrt {14} c_1\right ) e^{\left (\sqrt {14}-1\right ) x}-\frac {11}{169} \\ \end{align*}