61.7.1 problem Problem 3(a)

Internal problem ID [15393]
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 : Thursday, October 02, 2025 at 10:12:31 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*}

✓ Maple. Time used: 0.198 (sec). Leaf size: 65

ode:=[diff(x(t),t) = -4*x(t)+9*y(t)+12*exp(-t), diff(y(t),t) = -5*x(t)+2*y(t)]; 
dsolve(ode);
 

\begin{align*} x \left (t \right ) &= \frac {{\mathrm e}^{-t} \left (6 \sin \left (6 t \right ) c_1 +3 \sin \left (6 t \right ) c_2 +3 \cos \left (6 t \right ) c_1 -6 \cos \left (6 t \right ) c_2 -5\right )}{5} \\ y \left (t \right ) &= \frac {{\mathrm e}^{-t} \left (-5+3 \sin \left (6 t \right ) c_2 +3 \cos \left (6 t \right ) c_1 \right )}{3} \\ \end{align*}

✓ Mathematica. Time used: 0.035 (sec). Leaf size: 187

ode={D[x[t],t]==-4*x[t]+9*y[t]+12*Exp[-t],D[y[t],t]==-5*x[t]+2*y[t]}; 
ic={}; 
DSolve[{ode,ic},{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*}

✓ Sympy. Time used: 0.110 (sec). Leaf size: 109

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(4*x(t) - 9*y(t) + Derivative(x(t), t) - 12*exp(-t),0),Eq(5*x(t) - 2*y(t) + Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = - \left (\frac {3 C_{1}}{5} - \frac {6 C_{2}}{5}\right ) e^{- t} \sin {\left (6 t \right )} + \left (\frac {6 C_{1}}{5} + \frac {3 C_{2}}{5}\right ) e^{- t} \cos {\left (6 t \right )} - e^{- t} \sin ^{2}{\left (6 t \right )} - e^{- t} \cos ^{2}{\left (6 t \right )}, \ y{\left (t \right )} = - C_{1} e^{- t} \sin {\left (6 t \right )} + C_{2} e^{- t} \cos {\left (6 t \right )} - \frac {5 e^{- t} \sin ^{2}{\left (6 t \right )}}{3} - \frac {5 e^{- t} \cos ^{2}{\left (6 t \right )}}{3}\right ] \]