67.7.3 problem Problem 3(c)

Internal problem ID [14114]
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(c)
Date solved : Tuesday, January 28, 2025 at 06:14:39 AM
CAS classification : system_of_ODEs

\begin{align*} x^{\prime }\left (t \right )&=-7 x \left (t \right )+10 y+18 \,{\mathrm e}^{t}\\ y^{\prime }&=-10 x \left (t \right )+9 y+37 \end{align*}

✓ Solution by Maple

Time used: 0.108 (sec). Leaf size: 80

dsolve([diff(x(t),t)=-7*x(t)+10*y(t)+18*exp(t),diff(y(t),t)=-10*x(t)+9*y(t)+37],singsol=all)
 

\begin{align*} x \left (t \right ) &= 10+\frac {{\mathrm e}^{t} \left (-20+4 \cos \left (6 t \right ) c_{1} -3 \cos \left (6 t \right ) c_{2} +3 \sin \left (6 t \right ) c_{1} +4 \sin \left (6 t \right ) c_{2} -20 \cos \left (6 t \right )-15 \sin \left (6 t \right )\right )}{5} \\ y &= 7+{\mathrm e}^{t} \left (-5+\cos \left (6 t \right ) c_{1} +\sin \left (6 t \right ) c_{2} -5 \cos \left (6 t \right )\right ) \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.492 (sec). Leaf size: 260

DSolve[{D[x[t],t]==-7*x[t]+10*y[t]+18*Exp[t],D[y[t],t]==-10*x[t]+9*y[t]+37},{x[t],y[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} x(t)\to \frac {1}{3} e^t \left ((3 \cos (6 t)-4 \sin (6 t)) \int _1^t\left (18 \cos (6 K[1])+\frac {1}{3} \left (72-185 e^{-K[1]}\right ) \sin (6 K[1])\right )dK[1]+5 \sin (6 t) \int _1^t\left (\frac {37}{3} e^{-K[2]} (3 \cos (6 K[2])-4 \sin (6 K[2]))+30 \sin (6 K[2])\right )dK[2]+3 c_1 \cos (6 t)-4 c_1 \sin (6 t)+5 c_2 \sin (6 t)\right ) \\ y(t)\to \frac {1}{3} e^t \left (-5 \sin (6 t) \int _1^t\left (18 \cos (6 K[1])+\frac {1}{3} \left (72-185 e^{-K[1]}\right ) \sin (6 K[1])\right )dK[1]+(4 \sin (6 t)+3 \cos (6 t)) \int _1^t\left (\frac {37}{3} e^{-K[2]} (3 \cos (6 K[2])-4 \sin (6 K[2]))+30 \sin (6 K[2])\right )dK[2]+3 c_2 \cos (6 t)-5 c_1 \sin (6 t)+4 c_2 \sin (6 t)\right ) \\ \end{align*}