Internal problem ID [12081]
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 (i).
ODE order: 1.
ODE degree: 1.
Solve \begin {align*} x^{\prime }\left (t \right )&=4 x \left (t \right )-y \left (t \right )\\ y^{\prime }\left (t \right )&=2 x \left (t \right )+y \left (t \right )+t^{2} \end {align*}
With initial conditions \[ [x \left (0\right ) = 0, y \left (0\right ) = 1] \]
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 52
dsolve([diff(x(t),t) = 4*x(t)-y(t), diff(y(t),t) = 2*x(t)+y(t)+t^2, x(0) = 0, y(0) = 1], singsol=all)
\begin{align*} x \left (t \right ) &= -\frac {29 \,{\mathrm e}^{3 t}}{27}+\frac {5 \,{\mathrm e}^{2 t}}{4}-\frac {t^{2}}{6}-\frac {5 t}{18}-\frac {19}{108} \\ y \left (t \right ) &= -\frac {29 \,{\mathrm e}^{3 t}}{27}+\frac {5 \,{\mathrm e}^{2 t}}{2}-\frac {7 t}{9}-\frac {23}{54}-\frac {2 t^{2}}{3} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.128 (sec). Leaf size: 64
DSolve[{x'[t]==4*x[t]-y[t],y'[t]==2*x[t]+y[t]+t^2},{x[0]==0,y[0]==1},{x[t],y[t]},t,IncludeSingularSolutions -> True]
\begin{align*} x(t)\to \frac {1}{108} \left (-18 t^2-30 t+135 e^{2 t}-116 e^{3 t}-19\right ) \\ y(t)\to \frac {1}{54} \left (-36 t^2-42 t+135 e^{2 t}-58 e^{3 t}-23\right ) \\ \end{align*}