82.42.3 problem Ex. 3
Internal
problem
ID
[18888]
Book
:
Introductory
Course
On
Differential
Equations
by
Daniel
A
Murray.
Longmans
Green
and
Co.
NY.
1924
Section
:
Chapter
VIII.
Exact
differential
equations,
and
equations
of
particular
forms.
Integration
in
series.
problems
at
page
97
Problem
number
:
Ex.
3
Date
solved
:
Monday, March 31, 2025 at 06:18:35 PM
CAS
classification
:
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]
\begin{align*} y^{\prime \prime }+\frac {a^{2}}{y^{2}}&=0 \end{align*}
✓ Maple. Time used: 0.032 (sec). Leaf size: 269
ode:=diff(diff(y(x),x),x)+a^2/y(x)^2 = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= \frac {c_1 \left (-2 a^{2} c_1 +{\mathrm e}^{-\operatorname {RootOf}\left (\operatorname {csgn}\left (\frac {1}{c_1}\right ) c_1^{4} a^{4}+2 \textit {\_Z} \,c_1^{3} a^{2} {\mathrm e}^{\textit {\_Z}}-{\mathrm e}^{2 \textit {\_Z}} \operatorname {csgn}\left (\frac {1}{c_1}\right ) c_1^{2}-2 \,{\mathrm e}^{\textit {\_Z}} \operatorname {csgn}\left (\frac {1}{c_1}\right ) c_2 -2 \,{\mathrm e}^{\textit {\_Z}} \operatorname {csgn}\left (\frac {1}{c_1}\right ) x \right )} c_1^{2} a^{4}+{\mathrm e}^{\operatorname {RootOf}\left (\operatorname {csgn}\left (\frac {1}{c_1}\right ) c_1^{4} a^{4}+2 \textit {\_Z} \,c_1^{3} a^{2} {\mathrm e}^{\textit {\_Z}}-{\mathrm e}^{2 \textit {\_Z}} \operatorname {csgn}\left (\frac {1}{c_1}\right ) c_1^{2}-2 \,{\mathrm e}^{\textit {\_Z}} \operatorname {csgn}\left (\frac {1}{c_1}\right ) c_2 -2 \,{\mathrm e}^{\textit {\_Z}} \operatorname {csgn}\left (\frac {1}{c_1}\right ) x \right )}\right )}{2} \\
y &= \frac {c_1 \left ({\mathrm e}^{\operatorname {RootOf}\left (\operatorname {csgn}\left (\frac {1}{c_1}\right ) c_1^{4} a^{4}+2 \textit {\_Z} \,c_1^{3} a^{2} {\mathrm e}^{\textit {\_Z}}-{\mathrm e}^{2 \textit {\_Z}} \operatorname {csgn}\left (\frac {1}{c_1}\right ) c_1^{2}+2 \,{\mathrm e}^{\textit {\_Z}} \operatorname {csgn}\left (\frac {1}{c_1}\right ) c_2 +2 \,{\mathrm e}^{\textit {\_Z}} \operatorname {csgn}\left (\frac {1}{c_1}\right ) x \right )}-2 a^{2} c_1 +{\mathrm e}^{-\operatorname {RootOf}\left (\operatorname {csgn}\left (\frac {1}{c_1}\right ) c_1^{4} a^{4}+2 \textit {\_Z} \,c_1^{3} a^{2} {\mathrm e}^{\textit {\_Z}}-{\mathrm e}^{2 \textit {\_Z}} \operatorname {csgn}\left (\frac {1}{c_1}\right ) c_1^{2}+2 \,{\mathrm e}^{\textit {\_Z}} \operatorname {csgn}\left (\frac {1}{c_1}\right ) c_2 +2 \,{\mathrm e}^{\textit {\_Z}} \operatorname {csgn}\left (\frac {1}{c_1}\right ) x \right )} c_1^{2} a^{4}\right )}{2} \\
\end{align*}
✓ Mathematica. Time used: 0.169 (sec). Leaf size: 71
ode=D[y[x],{x,2}]+a^2/y[x]^2==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\[
\text {Solve}\left [\left (\frac {y(x) \sqrt {\frac {2 a^2}{y(x)}+c_1}}{c_1}-\frac {2 a^2 \text {arctanh}\left (\frac {\sqrt {\frac {2 a^2}{y(x)}+c_1}}{\sqrt {c_1}}\right )}{c_1{}^{3/2}}\right ){}^2=(x+c_2){}^2,y(x)\right ]
\]
✓ Sympy. Time used: 7.221 (sec). Leaf size: 190
from sympy import *
x = symbols("x")
a = symbols("a")
y = Function("y")
ode = Eq(a**2/y(x)**2 + Derivative(y(x), (x, 2)),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ - x + \frac {\sqrt {2} y^{\frac {3}{2}}{\left (x \right )}}{2 a \sqrt {\frac {C_{1} y{\left (x \right )}}{2 a^{2}} + 1}} + \frac {\sqrt {2} a \sqrt {y{\left (x \right )}}}{C_{1} \sqrt {\frac {C_{1} y{\left (x \right )}}{2 a^{2}} + 1}} - \frac {2 a^{2} \operatorname {asinh}{\left (\frac {\sqrt {2} \sqrt {C_{1}} \sqrt {y{\left (x \right )}}}{2 a} \right )}}{C_{1}^{\frac {3}{2}}} = C_{2}, \ x + \frac {\sqrt {2} y^{\frac {3}{2}}{\left (x \right )}}{2 a \sqrt {\frac {C_{1} y{\left (x \right )}}{2 a^{2}} + 1}} + \frac {\sqrt {2} a \sqrt {y{\left (x \right )}}}{C_{1} \sqrt {\frac {C_{1} y{\left (x \right )}}{2 a^{2}} + 1}} - \frac {2 a^{2} \operatorname {asinh}{\left (\frac {\sqrt {2} \sqrt {C_{1}} \sqrt {y{\left (x \right )}}}{2 a} \right )}}{C_{1}^{\frac {3}{2}}} = C_{2}\right ]
\]