64.16.3 problem 3

Internal problem ID [13606]
Book : Differential Equations by Shepley L. Ross. Third edition. John Willey. New Delhi. 2004.
Section : Chapter 7, Systems of linear differential equations. Section 7.1. Exercises page 277
Problem number : 3
Date solved : Tuesday, January 28, 2025 at 05:53:48 AM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.048 (sec). Leaf size: 36

dsolve([diff(x(t),t)+diff(y(t),t)-x(t)-3*y(t)=exp(t),diff(x(t),t)+diff(y(t),t)+x(t)=exp(3*t)],singsol=all)
 

\begin{align*} x \left (t \right ) &= \frac {{\mathrm e}^{t}}{4}+{\mathrm e}^{-3 t} c_{1} \\ y \left (t \right ) &= \frac {{\mathrm e}^{3 t}}{3}-\frac {{\mathrm e}^{t}}{2}-\frac {2 \,{\mathrm e}^{-3 t} c_{1}}{3} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.173 (sec). Leaf size: 55

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

\begin{align*} x(t)\to \frac {e^t}{4}+\frac {3}{16} c_1 e^{-3 t} \\ y(t)\to -\frac {e^t}{2}+\frac {e^{3 t}}{3}-\frac {1}{8} c_1 e^{-3 t} \\ \end{align*}