82.4.22 problem 28-22

Internal problem ID [21840]
Book : The Differential Equations Problem Solver. VOL. II. M. Fogiel director. REA, NY. 1978. ISBN 78-63609
Section : Chapter 28. Laplace transforms. Page 850
Problem number : 28-22
Date solved : Thursday, October 02, 2025 at 08:02:45 PM
CAS classification : system_of_ODEs

\begin{align*} y^{\prime }+z \left (t \right )&=t\\ z^{\prime }\left (t \right )+4 y&=0 \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.045 (sec). Leaf size: 35

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

\begin{align*} y \left (t \right ) &= \frac {7 \,{\mathrm e}^{2 t}}{8}+\frac {3 \,{\mathrm e}^{-2 t}}{8}-\frac {1}{4} \\ z \left (t \right ) &= -\frac {7 \,{\mathrm e}^{2 t}}{4}+\frac {3 \,{\mathrm e}^{-2 t}}{4}+t \\ \end{align*}

✓ Mathematica. Time used: 0.076 (sec). Leaf size: 48

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

\begin{align*} y(t)&\to \frac {1}{8} \left (3 e^{-2 t}+7 e^{2 t}-2\right )\\ z(t)&\to t+\frac {3 e^{-2 t}}{4}-\frac {7 e^{2 t}}{4} \end{align*}

✓ Sympy. Time used: 0.171 (sec). Leaf size: 42

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

\[ \left [ z{\left (t \right )} = t - \frac {7 e^{2 t}}{4} + \frac {3 e^{- 2 t}}{4}, \ y{\left (t \right )} = \frac {7 e^{2 t}}{8} - \frac {1}{4} + \frac {3 e^{- 2 t}}{8}\right ] \]