Internal problem ID [12376]
Book: APPLIED DIFFERENTIAL EQUATIONS The Primary Course by Vladimir A.
Dobrushkin. CRC Press 2015
Section: Chapter 8.3 Systems of Linear Differential Equations (Variation of Parameters).
Problems page 514
Problem number: Problem 3(a).
ODE order: 1.
ODE degree: 1.
Solve \begin {align*} x^{\prime }\left (t \right )&=-4 x \left (t \right )+9 y+12 \,{\mathrm e}^{-t}\\ y^{\prime }&=-5 x \left (t \right )+2 y \end {align*}
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 66
dsolve([diff(x(t),t)=-4*x(t)+9*y(t)+12*exp(-t),diff(y(t),t)=-5*x(t)+2*y(t)],singsol=all)
\begin{align*} x \left (t \right ) &= \frac {{\mathrm e}^{-t} \left (6 \sin \left (6 t \right ) c_{1} +3 \sin \left (6 t \right ) c_{2} +3 \cos \left (6 t \right ) c_{1} -6 \cos \left (6 t \right ) c_{2} -5\right )}{5} \\ y \left (t \right ) &= \frac {{\mathrm e}^{-t} \left (-5+3 \sin \left (6 t \right ) c_{2} +3 \cos \left (6 t \right ) c_{1} \right )}{3} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.095 (sec). Leaf size: 73
DSolve[{x'[t]==-4*x[t]+9*y[t]+12*Exp[-t],y'[t]==-5*x[t]+2*y[t]},{x[t],y[t]},t,IncludeSingularSolutions -> True]
\begin{align*} x(t)\to \frac {1}{2} e^{-t} (2 c_1 \cos (6 t)-(c_1-3 c_2) \sin (6 t)-2) \\ y(t)\to \frac {1}{6} e^{-t} (6 c_2 \cos (6 t)+(3 c_2-5 c_1) \sin (6 t)-10) \\ \end{align*}