82.42.4 problem Ex. 4
Internal
problem
ID
[18960]
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.
4
Date
solved
:
Tuesday, January 28, 2025 at 12:39:39 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*}
✓ Solution by Maple
Time used: 0.020 (sec). Leaf size: 385
dsolve(diff(y(x),x$2)-a^2/y(x)^2=0,y(x), singsol=all)
\begin{align*}
y \left (x \right ) &= \frac {c_{1} \left (c_{1}^{2} a^{4}+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 )}+{\mathrm e}^{2 \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 ) {\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} \\
y \left (x \right ) &= \frac {c_{1} \left (c_{1}^{2} a^{4}+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 )}+{\mathrm e}^{2 \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 ) {\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} \\
\end{align*}
✓ Solution by Mathematica
Time used: 0.173 (sec). Leaf size: 71
DSolve[D[y[x],{x,2}]-a^2/y[x]^2==0,y[x],x,IncludeSingularSolutions -> True]
\[
\text {Solve}\left [\left (\frac {2 a^2 \text {arctanh}\left (\frac {\sqrt {-\frac {2 a^2}{y(x)}+c_1}}{\sqrt {c_1}}\right )}{c_1{}^{3/2}}+\frac {y(x) \sqrt {-\frac {2 a^2}{y(x)}+c_1}}{c_1}\right ){}^2=(x+c_2){}^2,y(x)\right ]
\]