85.91.1 problem 8 (a)

Internal problem ID [23047]
Book : Applied Differential Equations. By Murray R. Spiegel. 3rd edition. 1980. Pearson. ISBN 978-0130400970
Section : Chapter 11. Matrix eigenvalue methods for systems of linear differential equations. A Exercises at page 509
Problem number : 8 (a)
Date solved : Thursday, October 02, 2025 at 09:17:49 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )+3 x \left (t \right )-2 y \left (t \right )&={\mathrm e}^{-t}\\ \frac {d}{d t}y \left (t \right )-x \left (t \right )+4 y \left (t \right )&=\sin \left (2 t \right ) \end{align*}

✓ Maple. Time used: 0.152 (sec). Leaf size: 71

ode:=[diff(x(t),t)+3*x(t)-2*y(t) = exp(-t), diff(y(t),t)-x(t)+4*y(t) = sin(2*t)]; 
dsolve(ode);
 

\begin{align*} x \left (t \right ) &= {\mathrm e}^{-5 t} c_2 +{\mathrm e}^{-2 t} c_1 -\frac {7 \cos \left (2 t \right )}{58}+\frac {3 \sin \left (2 t \right )}{58}+\frac {3 \,{\mathrm e}^{-t}}{4} \\ y \left (t \right ) &= -{\mathrm e}^{-5 t} c_2 +\frac {{\mathrm e}^{-2 t} c_1}{2}+\frac {23 \sin \left (2 t \right )}{116}-\frac {15 \cos \left (2 t \right )}{116}+\frac {{\mathrm e}^{-t}}{4} \\ \end{align*}

✓ Mathematica. Time used: 0.452 (sec). Leaf size: 115

ode={D[x[t],{t,1}]+3*x[t]-2*y[t]==Exp[-t], D[y[t],{t,1}]-x[t]+4*y[t]==Sin[2*t]}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to \frac {3}{58} \sin (2 t)-\frac {7}{58} \cos (2 t)+\frac {1}{12} e^{-5 t} \left (9 e^{4 t}+8 (c_1+c_2) e^{3 t}+4 (c_1-2 c_2)\right )\\ y(t)&\to \frac {1}{348} \left (69 \sin (2 t)-45 \cos (2 t)+29 e^{-5 t} \left (3 e^{4 t}+4 (c_1+c_2) e^{3 t}-4 c_1+8 c_2\right )\right ) \end{align*}

✓ Sympy. Time used: 2.099 (sec). Leaf size: 80

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(3*x(t) - 2*y(t) + Derivative(x(t), t) - exp(-t),0),Eq(-x(t) + 4*y(t) - sin(2*t) + Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = 2 C_{1} e^{- 2 t} - C_{2} e^{- 5 t} + \frac {3 \sin {\left (2 t \right )}}{58} - \frac {7 \cos {\left (2 t \right )}}{58} + \frac {3 e^{- t}}{4}, \ y{\left (t \right )} = C_{1} e^{- 2 t} + C_{2} e^{- 5 t} + \frac {23 \sin {\left (2 t \right )}}{116} - \frac {15 \cos {\left (2 t \right )}}{116} + \frac {e^{- t}}{4}\right ] \]