85.88.15 problem 2 (f)

Internal problem ID [23042]
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 (f)
Date solved : Thursday, October 02, 2025 at 09:17:44 PM
CAS classification : system_of_ODEs

\begin{align*} 3 x \left (t \right )-\frac {d}{d t}y \left (t \right )-2 y \left (t \right )&=8 t\\ \frac {d}{d t}x \left (t \right )-2 x \left (t \right )+y \left (t \right )&=16 \,{\mathrm e}^{-t} \end{align*}

✓ Maple. Time used: 0.083 (sec). Leaf size: 63

ode:=[3*x(t)-diff(y(t),t)-2*y(t) = 8*t, diff(x(t),t)-2*x(t)+y(t) = 16*exp(-t)]; 
dsolve(ode);
 

\begin{align*} x \left (t \right ) &= \frac {{\mathrm e}^{-t} c_1}{3}-8 \,{\mathrm e}^{-t} t +{\mathrm e}^{t} c_2 -12 \,{\mathrm e}^{-t}-8 t \\ y \left (t \right ) &= {\mathrm e}^{-t} c_1 -24 \,{\mathrm e}^{-t} t +{\mathrm e}^{t} c_2 -12 \,{\mathrm e}^{-t}-16 t +8 \\ \end{align*}

✓ Mathematica. Time used: 0.139 (sec). Leaf size: 139

ode={3*x[t]-D[y[t],t]-2*y[t]==8*t, D[x[t],t]-2*x[t]+y[t]==16*Exp[-t]}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to \frac {1}{2} e^{-t} \left (-8 \left (2 e^t+3 e^{2 t}-1\right ) t+12 \left (e^{2 t}-1\right ) \log \left (e^{2 t}\right )+3 c_1 e^{2 t}-c_2 e^{2 t}-24-c_1+c_2\right )\\ y(t)&\to \frac {1}{2} e^{-t} \left (e^t (16-32 t)+12 \left (e^{2 t}-3\right ) \log \left (e^{2 t}\right )-e^{2 t} (24 t-3 c_1+c_2)+3 (8 t-8-c_1+c_2)\right ) \end{align*}

✓ Sympy. Time used: 0.133 (sec). Leaf size: 51

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(-8*t + 3*x(t) - 2*y(t) - Derivative(y(t), t),0),Eq(-2*x(t) + y(t) + Derivative(x(t), t) - 16*exp(-t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = C_{2} e^{t} - 8 t - 8 t e^{- t} + \left (\frac {C_{1}}{3} - 12\right ) e^{- t}, \ y{\left (t \right )} = C_{2} e^{t} - 16 t - 24 t e^{- t} + \left (C_{1} - 12\right ) e^{- t} + 8\right ] \]