67.7.2 problem Problem 3(b)

Internal problem ID [14113]
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)
Date solved : Tuesday, January 28, 2025 at 06:14:37 AM
CAS classification : system_of_ODEs

\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: 1.092 (sec). Leaf size: 81

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+\cos \left (6 t \right ) c_{1} -\cos \left (6 t \right ) c_{2} +\sin \left (6 t \right ) c_{1} +\sin \left (6 t \right ) c_{2} -2 \cos \left (6 t \right )-2 \sin \left (6 t \right )\right )}{2} \\ y &= 7+{\mathrm e}^{-t} \left (-2+\cos \left (6 t \right ) c_{1} +\sin \left (6 t \right ) c_{2} -2 \cos \left (6 t \right )\right ) \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.123 (sec). Leaf size: 227

DSolve[{D[x[t],t]==-7*x[t]+6*y[t]+6*Exp[-t],D[y[t],t]==-12*x[t]+5*y[t]+37},{x[t],y[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} x(t)\to e^{-t} \left ((\cos (6 t)-\sin (6 t)) \int _1^t\left (6 \cos (6 K[1])-\left (-6+37 e^{K[1]}\right ) \sin (6 K[1])\right )dK[1]+\sin (6 t) \int _1^t\left (37 e^{K[2]} (\cos (6 K[2])-\sin (6 K[2]))+12 \sin (6 K[2])\right )dK[2]+c_1 \cos (6 t)-c_1 \sin (6 t)+c_2 \sin (6 t)\right ) \\ y(t)\to e^{-t} \left (-2 \sin (6 t) \int _1^t\left (6 \cos (6 K[1])-\left (-6+37 e^{K[1]}\right ) \sin (6 K[1])\right )dK[1]+(\sin (6 t)+\cos (6 t)) \int _1^t\left (37 e^{K[2]} (\cos (6 K[2])-\sin (6 K[2]))+12 \sin (6 K[2])\right )dK[2]+c_2 \cos (6 t)-2 c_1 \sin (6 t)+c_2 \sin (6 t)\right ) \\ \end{align*}