84.40.3 problem 27.8

Internal problem ID [22378]
Book : Schaums outline series. Differential Equations By Richard Bronson. 1973. McGraw-Hill Inc. ISBN 0-07-008009-7
Section : Chapter 27. Solutions of systems of linear differential equations with constant coefficients by Laplace transform. Supplementary problems
Problem number : 27.8
Date solved : Thursday, October 02, 2025 at 08:38:04 PM
CAS classification : system_of_ODEs

\begin{align*} w^{\prime }\left (t \right )-w \left (t \right )-2 y&=1\\ y^{\prime }-4 w \left (t \right )-3 y&=-1 \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&=2 \\ w \left (0\right )&=1 \\ \end{align*}

✓ Maple. Time used: 0.045 (sec). Leaf size: 31

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

\begin{align*} w \left (t \right ) &= -{\mathrm e}^{-t}+{\mathrm e}^{5 t}+1 \\ y \left (t \right ) &= {\mathrm e}^{-t}+2 \,{\mathrm e}^{5 t}-1 \\ \end{align*}

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 36

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

\begin{align*} w(t)&\to -e^{-t}+e^{5 t}+1\\ y(t)&\to e^{-t}+2 e^{5 t}-1 \end{align*}

✓ Sympy. Time used: 0.095 (sec). Leaf size: 29

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

\[ \left [ y{\left (t \right )} = 2 e^{5 t} - 1 + e^{- t}, \ w{\left (t \right )} = e^{5 t} + 1 - e^{- t}\right ] \]