84.41.5 problem 31.6

Internal problem ID [22387]
Book : Schaums outline series. Differential Equations By Richard Bronson. 1973. McGraw-Hill Inc. ISBN 0-07-008009-7
Section : Chapter 31. Solutions of linear systems with constant coefficients. Solved problems. Page 194
Problem number : 31.6
Date solved : Thursday, October 02, 2025 at 08:38:08 PM
CAS classification : system_of_ODEs

\begin{align*} x^{\prime }\left (t \right )&=y\\ y^{\prime }&=-2 y-5 z \left (t \right )+3\\ z^{\prime }\left (t \right )&=y+2 z \left (t \right ) \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.058 (sec). Leaf size: 43

ode:=[diff(x(t),t) = y(t), diff(y(t),t) = -2*y(t)-5*z(t)+3, diff(z(t),t) = y(t)+2*z(t)]; 
ic:=[x(0) = 0, y(0) = 0, z(0) = 1]; 
dsolve([ode,op(ic)]);
 

\begin{align*} x \left (t \right ) &= -6 t +6 \sin \left (t \right )+2 \cos \left (t \right )-2 \\ y \left (t \right ) &= -2 \sin \left (t \right )+6 \cos \left (t \right )-6 \\ z \left (t \right ) &= -2 \cos \left (t \right )+2 \sin \left (t \right )+3 \\ \end{align*}

✗ Mathematica

ode={D[x[t],t]==y[t],D[y[t],t]==-2*y[t]-5*z[t]+3,D[z[t],t]==y[t]+2*z[t]}; 
ic={x[0]==0,y[0]==0,z[0]==1}; 
DSolve[{ode,ic},{x[t],y[t],z[t]},t,IncludeSingularSolutions->True]
 

Not solved

✓ Sympy. Time used: 0.138 (sec). Leaf size: 75

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
z = Function("z") 
ode=[Eq(-y(t) + Derivative(x(t), t),0),Eq(2*y(t) + 5*z(t) + Derivative(y(t), t) - 3,0),Eq(-y(t) - 2*z(t) + Derivative(z(t), t),0)] 
ics = {x(0): 0, y(0): 0, z(0): 0} 
dsolve(ode,func=[x(t),y(t),z(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = - 6 t + 3 \sin ^{2}{\left (t \right )} + 6 \sin {\left (t \right )} + 3 \cos ^{2}{\left (t \right )} - 3 \cos {\left (t \right )}, \ y{\left (t \right )} = - 6 \sin ^{2}{\left (t \right )} + 3 \sin {\left (t \right )} - 6 \cos ^{2}{\left (t \right )} + 6 \cos {\left (t \right )}, \ z{\left (t \right )} = 3 \sin ^{2}{\left (t \right )} + 3 \cos ^{2}{\left (t \right )} - 3 \cos {\left (t \right )}\right ] \]