84.39.3 problem 27.3

Internal problem ID [22373]
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. Solved problems. Page 164
Problem number : 27.3
Date solved : Sunday, October 12, 2025 at 05:52:21 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d^{2}}{d t^{2}}y \left (t \right )+z \left (t \right )+y \left (t \right )&=0\\ \frac {d}{d t}z \left (t \right )+\frac {d}{d t}y \left (t \right )&=0 \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.024 (sec). Leaf size: 19

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

\begin{align*} y \left (t \right ) &= -t^{2}+1 \\ z \left (t \right ) &= t^{2}+1 \\ \end{align*}

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 20

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

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

✓ Sympy. Time used: 0.062 (sec). Leaf size: 14

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

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