88.12.22 problem 24
Internal
problem
ID
[24078]
Book
:
Elementary
Differential
Equations.
By
Lee
Roy
Wilcox
and
Herbert
J.
Curtis.
1961
first
edition.
International
texbook
company.
Scranton,
Penn.
USA.
CAT
number
61-15976
Section
:
Chapter
2.
Differential
equations
of
first
order.
Miscellaneous
Exercises
at
page
55
Problem
number
:
24
Date
solved
:
Thursday, October 02, 2025 at 09:57:28 PM
CAS
classification
:
system_of_ODEs
\begin{align*} y^{\prime }&=z \left (x \right )\\ z^{\prime }\left (x \right )&=w \left (x \right )\\ w^{\prime }\left (x \right )&=y \end{align*}
✓ Maple. Time used: 0.067 (sec). Leaf size: 175
ode:=[diff(y(x),x) = z(x), diff(z(x),x) = w(x), diff(w(x),x) = y(x)];
dsolve(ode);
\begin{align*}
w \left (x \right ) &= c_1 \,{\mathrm e}^{x}+c_2 \,{\mathrm e}^{-\frac {x}{2}} \sin \left (\frac {\sqrt {3}\, x}{2}\right )+c_3 \,{\mathrm e}^{-\frac {x}{2}} \cos \left (\frac {\sqrt {3}\, x}{2}\right ) \\
y \left (x \right ) &= c_1 \,{\mathrm e}^{x}-\frac {c_2 \,{\mathrm e}^{-\frac {x}{2}} \sin \left (\frac {\sqrt {3}\, x}{2}\right )}{2}+\frac {c_2 \,{\mathrm e}^{-\frac {x}{2}} \sqrt {3}\, \cos \left (\frac {\sqrt {3}\, x}{2}\right )}{2}-\frac {c_3 \,{\mathrm e}^{-\frac {x}{2}} \cos \left (\frac {\sqrt {3}\, x}{2}\right )}{2}-\frac {c_3 \,{\mathrm e}^{-\frac {x}{2}} \sqrt {3}\, \sin \left (\frac {\sqrt {3}\, x}{2}\right )}{2} \\
z \left (x \right ) &= c_1 \,{\mathrm e}^{x}-\frac {c_2 \,{\mathrm e}^{-\frac {x}{2}} \sin \left (\frac {\sqrt {3}\, x}{2}\right )}{2}-\frac {c_2 \,{\mathrm e}^{-\frac {x}{2}} \sqrt {3}\, \cos \left (\frac {\sqrt {3}\, x}{2}\right )}{2}-\frac {c_3 \,{\mathrm e}^{-\frac {x}{2}} \cos \left (\frac {\sqrt {3}\, x}{2}\right )}{2}+\frac {c_3 \,{\mathrm e}^{-\frac {x}{2}} \sqrt {3}\, \sin \left (\frac {\sqrt {3}\, x}{2}\right )}{2} \\
\end{align*}
✓ Mathematica. Time used: 0.021 (sec). Leaf size: 234
ode={D[y[x],x]==z[x],D[z[x],x]==w[x],D[w[x],x]==y[x]};
ic={};
DSolve[{ode,ic},{y[x],z[x],w[x]},x,IncludeSingularSolutions->True]
\begin{align*} w(x)&\to \frac {1}{3} e^{-x/2} \left ((c_1+c_2+c_3) e^{3 x/2}+(2 c_1-c_2-c_3) \cos \left (\frac {\sqrt {3} x}{2}\right )+\sqrt {3} (c_2-c_3) \sin \left (\frac {\sqrt {3} x}{2}\right )\right )\\ y(x)&\to \frac {1}{3} e^{-x/2} \left ((c_1+c_2+c_3) e^{3 x/2}-(c_1-2 c_2+c_3) \cos \left (\frac {\sqrt {3} x}{2}\right )-\sqrt {3} (c_1-c_3) \sin \left (\frac {\sqrt {3} x}{2}\right )\right )\\ z(x)&\to \frac {1}{3} e^{-x/2} \left ((c_1+c_2+c_3) e^{3 x/2}-(c_1+c_2-2 c_3) \cos \left (\frac {\sqrt {3} x}{2}\right )+\sqrt {3} (c_1-c_2) \sin \left (\frac {\sqrt {3} x}{2}\right )\right ) \end{align*}
✓ Sympy. Time used: 0.165 (sec). Leaf size: 163
from sympy import *
x = symbols("x")
y = Function("y")
z = Function("z")
w = Function("w")
ode=[Eq(-z(x) + Derivative(y(x), x),0),Eq(-w(x) + Derivative(z(x), x),0),Eq(-y(x) + Derivative(w(x), x),0)]
ics = {}
dsolve(ode,func=[y(x),z(x),w(x)],ics=ics)
\[
\left [ y{\left (x \right )} = C_{3} e^{x} - \left (\frac {C_{1}}{2} + \frac {\sqrt {3} C_{2}}{2}\right ) e^{- \frac {x}{2}} \cos {\left (\frac {\sqrt {3} x}{2} \right )} - \left (\frac {\sqrt {3} C_{1}}{2} - \frac {C_{2}}{2}\right ) e^{- \frac {x}{2}} \sin {\left (\frac {\sqrt {3} x}{2} \right )}, \ z{\left (x \right )} = C_{3} e^{x} - \left (\frac {C_{1}}{2} - \frac {\sqrt {3} C_{2}}{2}\right ) e^{- \frac {x}{2}} \cos {\left (\frac {\sqrt {3} x}{2} \right )} + \left (\frac {\sqrt {3} C_{1}}{2} + \frac {C_{2}}{2}\right ) e^{- \frac {x}{2}} \sin {\left (\frac {\sqrt {3} x}{2} \right )}, \ w{\left (x \right )} = C_{1} e^{- \frac {x}{2}} \cos {\left (\frac {\sqrt {3} x}{2} \right )} - C_{2} e^{- \frac {x}{2}} \sin {\left (\frac {\sqrt {3} x}{2} \right )} + C_{3} e^{x}\right ]
\]