67.7.1 problem Problem 3(a)

Internal problem ID [14112]
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)
Date solved : Tuesday, January 28, 2025 at 06:14:36 AM
CAS classification : system_of_ODEs

\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: 0.058 (sec). Leaf size: 65

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

✓ Solution by Mathematica

Time used: 0.046 (sec). Leaf size: 187

DSolve[{D[x[t],t]==-4*x[t]+9*y[t]+12*Exp[-t],D[y[t],t]==-5*x[t]+2*y[t]},{x[t],y[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} x(t)\to \frac {1}{2} e^{-t} \left (3 \sin (6 t) \int _1^t10 \sin (6 K[2])dK[2]+(2 \cos (6 t)-\sin (6 t)) \int _1^t6 (2 \cos (6 K[1])+\sin (6 K[1]))dK[1]+2 c_1 \cos (6 t)-c_1 \sin (6 t)+3 c_2 \sin (6 t)\right ) \\ y(t)\to \frac {1}{6} e^{-t} \left (-5 \sin (6 t) \int _1^t6 (2 \cos (6 K[1])+\sin (6 K[1]))dK[1]+3 (\sin (6 t)+2 \cos (6 t)) \int _1^t10 \sin (6 K[2])dK[2]+6 c_2 \cos (6 t)-5 c_1 \sin (6 t)+3 c_2 \sin (6 t)\right ) \\ \end{align*}