85.92.8 problem 2 (b)

Internal problem ID [23057]
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 528
Problem number : 2 (b)
Date solved : Thursday, October 02, 2025 at 09:18:24 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )+3 x \left (t \right )+4 y \left (t \right )&=8 \,{\mathrm e}^{t}\\ -x \left (t \right )+\frac {d}{d t}y \left (t \right )-y \left (t \right )&=0 \end{align*}

✓ Maple. Time used: 0.061 (sec). Leaf size: 42

ode:=[diff(x(t),t)+3*x(t)+4*y(t) = 8*exp(t), diff(y(t),t)-x(t)-y(t) = 0]; 
dsolve(ode);
 

\begin{align*} x \left (t \right ) &= -{\mathrm e}^{-t} \left (2 c_1 t -c_1 +2 c_2 \right ) \\ y \left (t \right ) &= {\mathrm e}^{-t} c_2 +{\mathrm e}^{-t} t c_1 +2 \,{\mathrm e}^{t} \\ \end{align*}

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 52

ode={D[x[t],{t,1}]+3*x[t]+4*y[t]==8*Exp[t], D[y[t],{t,1}]-x[t]-y[t]==0}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to e^{-t} (-2 c_1 t-4 c_2 t+c_1)\\ y(t)&\to e^{-t} \left (2 e^{2 t}+(c_1+2 c_2) t+c_2\right ) \end{align*}

✓ Sympy. Time used: 0.088 (sec). Leaf size: 39

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(3*x(t) + 4*y(t) - 8*exp(t) + Derivative(x(t), t),0),Eq(-x(t) - y(t) + Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = - 2 C_{2} t e^{- t} - \left (2 C_{1} - C_{2}\right ) e^{- t}, \ y{\left (t \right )} = C_{1} e^{- t} + C_{2} t e^{- t} + 2 e^{t}\right ] \]