Internal problem ID [11766]
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).
ODE order: 1.
ODE degree: 1.
Solve \begin {align*} x^{\prime }\left (t \right )&=x \left (t \right )+y \left (t \right )+{\mathrm e}^{-t}\\ y^{\prime }\left (t \right )&=4 x \left (t \right )-2 y \left (t \right )+{\mathrm e}^{2 t} \end {align*}
With initial conditions \[ [x \left (0\right ) = 1, y \left (0\right ) = -1] \]
✓ Solution by Maple
Time used: 0.141 (sec). Leaf size: 60
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],[x(t), y(t)], singsol=all)
\[ x \left (t \right ) = \frac {62 \,{\mathrm e}^{2 t}}{75}+\frac {{\mathrm e}^{2 t} t}{5}+\frac {17 \,{\mathrm e}^{-3 t}}{50}-\frac {{\mathrm e}^{-t}}{6} \] \[ y \left (t \right ) = \frac {77 \,{\mathrm e}^{2 t}}{75}-\frac {34 \,{\mathrm e}^{-3 t}}{25}+\frac {{\mathrm e}^{2 t} t}{5}-\frac {2 \,{\mathrm e}^{-t}}{3} \]
✓ Solution by Mathematica
Time used: 0.69 (sec). Leaf size: 67
DSolve[{x'[t]==x[t]+y[t]+Exp[-t],y'[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*}