65.14.6 problem 26.1 (vi)

Internal problem ID [13822]
Book : AN INTRODUCTION TO ORDINARY DIFFERENTIAL EQUATIONS by JAMES C. ROBINSON. Cambridge University Press 2004
Section : Chapter 26, Explicit solutions of coupled linear systems. Exercises page 257
Problem number : 26.1 (vi)
Date solved : Tuesday, January 28, 2025 at 06:04:46 AM
CAS classification : system_of_ODEs

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

With initial conditions

\begin{align*} x \left (0\right ) = 1\\ y \left (0\right ) = -1 \end{align*}

✓ Solution by Maple

Time used: 0.146 (sec). Leaf size: 59

dsolve([diff(x(t),t) = x(t)+y(t)+exp(-t), diff(y(t),t) = 4*x(t)-2*y(t)+exp(2*t), x(0) = 1, y(0) = -1], singsol=all)
 

\begin{align*} x \left (t \right ) &= \frac {17 \,{\mathrm e}^{-3 t}}{50}+\frac {62 \,{\mathrm e}^{2 t}}{75}+\frac {t \,{\mathrm e}^{2 t}}{5}-\frac {{\mathrm e}^{-t}}{6} \\ y \left (t \right ) &= \frac {77 \,{\mathrm e}^{2 t}}{75}-\frac {34 \,{\mathrm e}^{-3 t}}{25}+\frac {t \,{\mathrm e}^{2 t}}{5}-\frac {2 \,{\mathrm e}^{-t}}{3} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.681 (sec). Leaf size: 67

DSolve[{D[x[t],t]==x[t]+y[t]+Exp[-t],D[y[t],t]==4*x[t]-2*y[t]+Exp[2*t]},{x[0]==1,y[0]==-1},{x[t],y[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} x(t)\to \frac {1}{150} e^{-3 t} \left (2 e^{5 t} (15 t+62)-25 e^{2 t}+51\right ) \\ y(t)\to \frac {1}{75} e^{-3 t} \left (e^{5 t} (15 t+77)-50 e^{2 t}-102\right ) \\ \end{align*}