81.7.4 problem 5
Internal
problem
ID
[18701]
Book
:
A
short
course
on
differential
equations.
By
Donald
Francis
Campbell.
Maxmillan
company.
London.
1907
Section
:
Chapter
VI.
Certain
particular
forms
of
equations.
Exercises
at
page
74
Problem
number
:
5
Date
solved
:
Tuesday, January 28, 2025 at 12:10:47 PM
CAS
classification
:
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]
\begin{align*} y^{\prime \prime }&=\frac {1}{y^{2}} \end{align*}
✓ Solution by Maple
Time used: 0.020 (sec). Leaf size: 335
dsolve(diff(y(x),x$2)=1/y(x)^2,y(x), singsol=all)
\begin{align*}
y \left (x \right ) &= \frac {c_{1} \left ({\mathrm e}^{2 \operatorname {RootOf}\left (-{\mathrm e}^{2 \textit {\_Z}} \operatorname {csgn}\left (\frac {1}{c_{1}}\right ) c_{1}^{2}-2 \textit {\_Z} \,c_{1}^{3} {\mathrm e}^{\textit {\_Z}}+\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) c_{1}^{4}-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 \,{\mathrm e}^{\operatorname {RootOf}\left (-{\mathrm e}^{2 \textit {\_Z}} \operatorname {csgn}\left (\frac {1}{c_{1}}\right ) c_{1}^{2}-2 \textit {\_Z} \,c_{1}^{3} {\mathrm e}^{\textit {\_Z}}+\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) c_{1}^{4}-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} +c_{1}^{2}\right ) {\mathrm e}^{-\operatorname {RootOf}\left (-{\mathrm e}^{2 \textit {\_Z}} \operatorname {csgn}\left (\frac {1}{c_{1}}\right ) c_{1}^{2}-2 \textit {\_Z} \,c_{1}^{3} {\mathrm e}^{\textit {\_Z}}+\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) c_{1}^{4}-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 ({\mathrm e}^{2 \operatorname {RootOf}\left (-{\mathrm e}^{2 \textit {\_Z}} \operatorname {csgn}\left (\frac {1}{c_{1}}\right ) c_{1}^{2}-2 \textit {\_Z} \,c_{1}^{3} {\mathrm e}^{\textit {\_Z}}+\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) c_{1}^{4}+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 \,{\mathrm e}^{\operatorname {RootOf}\left (-{\mathrm e}^{2 \textit {\_Z}} \operatorname {csgn}\left (\frac {1}{c_{1}}\right ) c_{1}^{2}-2 \textit {\_Z} \,c_{1}^{3} {\mathrm e}^{\textit {\_Z}}+\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) c_{1}^{4}+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} +c_{1}^{2}\right ) {\mathrm e}^{-\operatorname {RootOf}\left (-{\mathrm e}^{2 \textit {\_Z}} \operatorname {csgn}\left (\frac {1}{c_{1}}\right ) c_{1}^{2}-2 \textit {\_Z} \,c_{1}^{3} {\mathrm e}^{\textit {\_Z}}+\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) c_{1}^{4}+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: 0.148 (sec). Leaf size: 62
DSolve[D[y[x],{x,2}]==1/y[x]^2,y[x],x,IncludeSingularSolutions -> True]
\[
\text {Solve}\left [\left (\frac {2 \text {arctanh}\left (\frac {\sqrt {-\frac {2}{y(x)}+c_1}}{\sqrt {c_1}}\right )}{c_1{}^{3/2}}+\frac {y(x) \sqrt {-\frac {2}{y(x)}+c_1}}{c_1}\right ){}^2=(x+c_2){}^2,y(x)\right ]
\]