85.86.3 problem 1 (c)

Internal problem ID [23020]
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 491
Problem number : 1 (c)
Date solved : Thursday, October 02, 2025 at 09:17:34 PM
CAS classification : system_of_ODEs

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

With initial conditions

\begin{align*} x \left (0\right )&=0 \\ y \left (0\right )&=1 \\ \end{align*}

✓ Maple. Time used: 0.064 (sec). Leaf size: 32

ode:=[diff(x(t),t)+3*diff(y(t),t)+y(t) = exp(t), diff(y(t),t)-x(t) = y(t)]; 
ic:=[x(0) = 0, y(0) = 1]; 
dsolve([ode,op(ic)]);
 

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

✓ Mathematica. Time used: 0.019 (sec). Leaf size: 35

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

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

✓ Sympy. Time used: 0.143 (sec). Leaf size: 32

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(y(t) - exp(t) + Derivative(x(t), t) + 3*Derivative(y(t), t),0),Eq(-x(t) - y(t) + Derivative(y(t), t),0)] 
ics = {x(0): 0, y(0): 1} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

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