84.40.1 problem 27.6

Internal problem ID [22376]
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 : 27.6
Date solved : Thursday, October 02, 2025 at 08:38:03 PM
CAS classification : system_of_ODEs

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

With initial conditions

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

✓ Maple. Time used: 0.049 (sec). Leaf size: 9

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

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

✓ Mathematica. Time used: 0.013 (sec). Leaf size: 10

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

\begin{align*} y(t)&\to 1\\ z(t)&\to t \end{align*}

✓ Sympy. Time used: 0.104 (sec). Leaf size: 27

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

\[ \left [ y{\left (t \right )} = \sin ^{2}{\left (t \right )} + \cos ^{2}{\left (t \right )}, \ z{\left (t \right )} = t \sin ^{2}{\left (t \right )} + t \cos ^{2}{\left (t \right )}\right ] \]