86.12.4 problem 6

Internal problem ID [23211]
Book : An introduction to Differential Equations. By Howard Frederick Cleaves. 1969. Oliver and Boyd publisher. ISBN 0050015044
Section : Chapter 10. The Laplace transform. Exercise 10c at page 156
Problem number : 6
Date solved : Thursday, October 02, 2025 at 09:24:25 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )+2 x \left (t \right )+\frac {d}{d t}y \left (t \right )+y \left (t \right )&={\mathrm e}^{-3 t}\\ 5 x \left (t \right )+\frac {d}{d t}y \left (t \right )+3 y \left (t \right )&=5 \,{\mathrm e}^{-t} \end{align*}

With initial conditions

\begin{align*} x \left (0\right )&=-1 \\ y \left (0\right )&=4 \\ \end{align*}

✓ Maple. Time used: 0.090 (sec). Leaf size: 35

ode:=[diff(x(t),t)+diff(y(t),t)+2*x(t)+y(t) = exp(-3*t), diff(y(t),t)+5*x(t)+3*y(t) = 5*exp(-t)]; 
ic:=[x(0) = -1, y(0) = 4]; 
dsolve([ode,op(ic)]);
 

\begin{align*} x \left (t \right ) &= \sin \left (t \right )-\cos \left (t \right ) \\ y \left (t \right ) &= -\sin \left (t \right )+2 \cos \left (t \right )+\frac {5 \,{\mathrm e}^{-t}}{2}-\frac {{\mathrm e}^{-3 t}}{2} \\ \end{align*}

✓ Mathematica. Time used: 0.127 (sec). Leaf size: 42

ode={D[x[t],{t,1}]+D[y[t],t]+2*x[t]+y[t]==Exp[-3*t],D[y[t],{t,1}]+5*x[t]+3*y[t]==5*Exp[-t]}; 
ic={x[0]==-1,y[0]==4}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to \sin (t)-\cos (t)\\ y(t)&\to \frac {1}{2} \left (-e^{-3 t}+5 e^{-t}-2 \sin (t)+4 \cos (t)\right ) \end{align*}

✓ Sympy. Time used: 0.535 (sec). Leaf size: 66

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(2*x(t) + y(t) + Derivative(x(t), t) + Derivative(y(t), t) - exp(-3*t),0),Eq(5*x(t) + 3*y(t) + Derivative(y(t), t) - 5*exp(-t),0)] 
ics = {x(0): -1, y(0): 4} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = \sin {\left (t \right )} - \cos {\left (t \right )}, \ y{\left (t \right )} = - \sin {\left (t \right )} + 2 \cos {\left (t \right )} + \frac {5 e^{- t} \sin ^{2}{\left (t \right )}}{2} + \frac {5 e^{- t} \cos ^{2}{\left (t \right )}}{2} - \frac {e^{- 3 t} \sin ^{2}{\left (t \right )}}{2} - \frac {e^{- 3 t} \cos ^{2}{\left (t \right )}}{2}\right ] \]