85.92.9 problem 2 (c)

Internal problem ID [23058]
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 (c)
Date solved : Thursday, October 02, 2025 at 09:18:25 PM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.078 (sec). Leaf size: 61

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

\begin{align*} x \left (t \right ) &= {\mathrm e}^{t} c_2 +\frac {{\mathrm e}^{-t} c_1}{3}-\frac {{\mathrm e}^{-t} t}{2}-\frac {3 \,{\mathrm e}^{-t}}{4}+t \\ y \left (t \right ) &= {\mathrm e}^{t} c_2 +{\mathrm e}^{-t} c_1 -\frac {3 \,{\mathrm e}^{-t} t}{2}-\frac {3 \,{\mathrm e}^{-t}}{4}+2 t -1 \\ \end{align*}

✓ Mathematica. Time used: 0.087 (sec). Leaf size: 145

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

\begin{align*} x(t)&\to \frac {1}{8} e^{-t} \left (3 \left (e^{2 t}-1\right ) \log \left (e^{2 t}\right )+2 \left (4 e^t t-3 e^{2 t} t+t+6 c_1 e^{2 t}-2 c_2 e^{2 t}-3-2 c_1+2 c_2\right )\right )\\ y(t)&\to \frac {1}{8} e^{-t} \left (3 \left (e^{2 t}-3\right ) \log \left (e^{2 t}\right )-2 \left (e^t (4-8 t)-3 t+e^{2 t} (3 t-6 c_1+2 c_2)+3+6 c_1-6 c_2\right )\right ) \end{align*}

✓ Sympy. Time used: 0.126 (sec). Leaf size: 54

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

\[ \left [ x{\left (t \right )} = C_{2} e^{t} + t - \frac {t e^{- t}}{2} + \left (\frac {C_{1}}{3} - \frac {3}{4}\right ) e^{- t}, \ y{\left (t \right )} = C_{2} e^{t} + 2 t - \frac {3 t e^{- t}}{2} + \left (C_{1} - \frac {3}{4}\right ) e^{- t} - 1\right ] \]