84.39.4 problem 27.4

Internal problem ID [22374]
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 : 27.4
Date solved : Sunday, October 12, 2025 at 05:52:21 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d^{2}}{d t^{2}}z \left (t \right )+\frac {d}{d t}y \left (t \right )&=\cos \left (t \right )\\ \frac {d^{2}}{d t^{2}}y \left (t \right )-z \left (t \right )&=\sin \left (t \right ) \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 \\ D\left (z \right )\left (0\right )&=-1 \\ \end{align*}

✓ Maple. Time used: 0.951 (sec). Leaf size: 18

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

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

✓ Mathematica. Time used: 0.872 (sec). Leaf size: 3745

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

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✓ Sympy. Time used: 3.138 (sec). Leaf size: 153

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

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