Internal problem ID [12377]
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(b).
ODE order: 1.
ODE degree: 1.
Solve \begin {align*} x^{\prime }\left (t \right )&=-7 x \left (t \right )+6 y+6 \,{\mathrm e}^{-t}\\ y^{\prime }&=-12 x \left (t \right )+5 y+37 \end {align*}
✓ Solution by Maple
Time used: 0.156 (sec). Leaf size: 82
dsolve([diff(x(t),t)=-7*x(t)+6*y(t)+6*exp(-t),diff(y(t),t)=-12*x(t)+5*y(t)+37],singsol=all)
\begin{align*} x \left (t \right ) &= 6+\frac {{\mathrm e}^{-t} \left (-2+\sin \left (6 t \right ) c_{1} +\sin \left (6 t \right ) c_{2} +\cos \left (6 t \right ) c_{1} -\cos \left (6 t \right ) c_{2} -2 \sin \left (6 t \right )-2 \cos \left (6 t \right )\right )}{2} \\ y \left (t \right ) &= 7+{\mathrm e}^{-t} \left (-2+\sin \left (6 t \right ) c_{2} +\cos \left (6 t \right ) c_{1} -2 \cos \left (6 t \right )\right ) \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.387 (sec). Leaf size: 72
DSolve[{x'[t]==-7*x[t]+6*y[t]+6*Exp[-t],y'[t]==-12*x[t]+5*y[t]+37},{x[t],y[t]},t,IncludeSingularSolutions -> True]
\begin{align*} x(t)\to e^{-t} \left (6 e^t+c_1 \cos (6 t)+(c_2-c_1) \sin (6 t)-1\right ) \\ y(t)\to e^{-t} \left (7 e^t+c_2 \cos (6 t)+(c_2-2 c_1) \sin (6 t)-2\right ) \\ \end{align*}