62.3.4 problem 6.3 (e)

Internal problem ID [15438]
Book : Differential Equations, Linear, Nonlinear, Ordinary, Partial. A.C. King, J.Billingham, S.R.Otto. Cambridge Univ. Press 2003
Section : Chapter 6. Laplace transforms. Problems page 172
Problem number : 6.3 (e)
Date solved : Thursday, October 02, 2025 at 10:14:10 AM
CAS classification : system_of_ODEs

\begin{align*} x^{\prime }\left (t \right )+y^{\prime }+x \left (t \right )&=0\\ x^{\prime }\left (t \right )-x \left (t \right )+2 y^{\prime }&={\mathrm e}^{-t} \end{align*}

With initial conditions

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

✗ Maple

ode:=[diff(x(t),t)+x(t)+diff(y(t),t) = 0, diff(x(t),t)-x(t)+2*diff(y(t),t) = exp(-t)]; 
ic:=[x(0) = 0, y(0) = 1]; 
dsolve([ode,op(ic)]);
 

\[ \text {No solution found} \]

✓ Mathematica. Time used: 0.074 (sec). Leaf size: 37

ode={D[x[t],t]+x[t]+D[y[t],t]==0,D[x[t],t]-x[t]+2*D[y[t],t]==Exp[-t]}; 
ic={x[0]==0,y[0]==1}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.122 (sec). Leaf size: 31

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(x(t) + Derivative(x(t), t) + Derivative(y(t), t),0),Eq(-x(t) + Derivative(x(t), t) + 2*Derivative(y(t), t) - exp(-t),0)] 
ics = {x(0): 0, y(0): 1} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

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