85.88.16 problem 2 (g)

Internal problem ID [23043]
Book : Applied Differential Equations. By Murray R. Spiegel. 3rd edition. 1980. Pearson. ISBN 978-0130400970
Section : Chapter 10. Systems of differential equations and their applications. A Exercises at page 499
Problem number : 2 (g)
Date solved : Thursday, October 02, 2025 at 09:17:46 PM
CAS classification : system_of_ODEs

\begin{align*} 2 \frac {d}{d t}x \left (t \right )-x \left (t \right )-\frac {d}{d t}y \left (t \right )+y \left (t \right )&=4 t \,{\mathrm e}^{-t}-3 \,{\mathrm e}^{-t}\\ \frac {d}{d t}x \left (t \right )+4 x \left (t \right )-2 \frac {d}{d t}y \left (t \right )-4 y \left (t \right )&=2 t \,{\mathrm e}^{-t}-6 \,{\mathrm e}^{-t} \end{align*}

✓ Maple. Time used: 0.079 (sec). Leaf size: 50

ode:=[2*diff(x(t),t)-x(t)-diff(y(t),t)+y(t) = 4*exp(-t)*t-3*exp(-t), diff(x(t),t)+4*x(t)-2*diff(y(t),t)-4*y(t) = 2*exp(-t)*t-6*exp(-t)]; 
dsolve(ode);
 

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

✓ Mathematica. Time used: 0.025 (sec). Leaf size: 73

ode={2*D[x[t],t]-x[t]-D[y[t],t]+y[t]==4*t*Exp[-t]-3*Exp[-t], D[x[t],t]+4*x[t]-2*D[y[t],t]-4*y[t]==2*t*Exp[-t]-6*Exp[-t]}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.121 (sec). Leaf size: 44

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(-4*t*exp(-t) - x(t) + y(t) + 2*Derivative(x(t), t) - Derivative(y(t), t) + 3*exp(-t),0),Eq(-2*t*exp(-t) + 4*x(t) - 4*y(t) + Derivative(x(t), t) - 2*Derivative(y(t), t) + 6*exp(-t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

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