85.88.13 problem 2 (c)

Internal problem ID [23040]
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 (c)
Date solved : Thursday, October 02, 2025 at 09:17:43 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )-x \left (t \right )+2 \frac {d}{d t}y \left (t \right )+7 y \left (t \right )&=3 t -15\\ 2 \frac {d}{d t}x \left (t \right )+\frac {d}{d t}y \left (t \right )+x \left (t \right )+5 y \left (t \right )&=9 t -7 \end{align*}

With initial conditions

\begin{align*} x \left (0\right )&=0 \\ y \left (0\right )&=-3 \\ \end{align*}

✓ Maple. Time used: 0.057 (sec). Leaf size: 41

ode:=[diff(x(t),t)-x(t)+2*diff(y(t),t)+7*y(t) = 3*t-15, 2*diff(x(t),t)+x(t)+diff(y(t),t)+5*y(t) = 9*t-7]; 
ic:=[x(0) = 0, y(0) = -3]; 
dsolve([ode,op(ic)]);
 

\begin{align*} x \left (t \right ) &= \frac {7 \,{\mathrm e}^{-2 t}}{12}+\frac {{\mathrm e}^{-2 t} t}{2}+4 t -\frac {7}{12} \\ y \left (t \right ) &= \frac {{\mathrm e}^{-2 t}}{12}+\frac {{\mathrm e}^{-2 t} t}{2}-\frac {37}{12}+t \\ \end{align*}

✓ Mathematica. Time used: 0.054 (sec). Leaf size: 58

ode={D[x[t],{t,1}]-x[t]+2*D[y[t],t]+7*y[t]==3*(t-5), 2*D[x[t],t]+x[t]+ D[y[t],{t,1}]+5*y[t]==9*t-7}; 
ic={x[0]==0,y[0]==-3}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to \frac {1}{12} e^{-2 t} \left (6 t+e^{2 t} (48 t-7)+7\right )\\ y(t)&\to \frac {1}{12} e^{-2 t} \left (6 t+e^{2 t} (12 t-37)+1\right ) \end{align*}

✓ Sympy. Time used: 0.246 (sec). Leaf size: 49

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(-3*t - x(t) + 7*y(t) + Derivative(x(t), t) + 2*Derivative(y(t), t) + 15,0),Eq(-9*t + x(t) + 5*y(t) + 2*Derivative(x(t), t) + Derivative(y(t), t) + 7,0)] 
ics = {x(0): 0, y(0): -3} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = 4 t + \frac {t e^{- 2 t}}{2} - \frac {7}{12} + \frac {7 e^{- 2 t}}{12}, \ y{\left (t \right )} = t + \frac {t e^{- 2 t}}{2} - \frac {37}{12} + \frac {e^{- 2 t}}{12}\right ] \]