Internal problem ID [12379]
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(d).
ODE order: 1.
ODE degree: 1.
Solve \begin {align*} x^{\prime }\left (t \right )&=-14 x \left (t \right )+39 y+78 \sinh \left (t \right )\\ y^{\prime }&=-6 x \left (t \right )+16 y+6 \cosh \left (t \right ) \end {align*}
✓ Solution by Maple
Time used: 0.25 (sec). Leaf size: 86
dsolve([diff(x(t),t)=-14*x(t)+39*y(t)+78*sinh(t),diff(y(t),t)=-6*x(t)+16*y(t)+6*cosh(t)],singsol=all)
\begin{align*} x \left (t \right ) &= {\mathrm e}^{t} \sin \left (3 t \right ) c_{2} +{\mathrm e}^{t} \cos \left (3 t \right ) c_{1} -52 \,{\mathrm e}^{t}+60 \,{\mathrm e}^{-t} \\ y \left (t \right ) &= \frac {5 \,{\mathrm e}^{t} \sin \left (3 t \right ) c_{2}}{13}+\frac {{\mathrm e}^{t} \cos \left (3 t \right ) c_{2}}{13}+\frac {5 \,{\mathrm e}^{t} \cos \left (3 t \right ) c_{1}}{13}-\frac {{\mathrm e}^{t} \sin \left (3 t \right ) c_{1}}{13}-20 \,{\mathrm e}^{t}+20 \,{\mathrm e}^{-t}-2 \sinh \left (t \right ) \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.623 (sec). Leaf size: 90
DSolve[{x'[t]==-14*x[t]+39*y[t]+78*Sinh[t],y'[t]==-6*x[t]+16*y[t]+6*Cosh[t]},{x[t],y[t]},t,IncludeSingularSolutions -> True]
\begin{align*} x(t)\to 60 e^{-t}-52 e^t+c_1 e^t \cos (3 t)-(5 c_1-13 c_2) e^t \sin (3 t) \\ y(t)\to 21 e^{-t}-21 e^t+c_2 e^t \cos (3 t)-(2 c_1-5 c_2) e^t \sin (3 t) \\ \end{align*}