60.7.188 problem 1779 (book 6.188)
Internal
problem
ID
[11777]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
6,
non-linear
second
order
Problem
number
:
1779
(book
6.188)
Date
solved
:
Monday, January 27, 2025 at 11:34:27 PM
CAS
classification
:
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]
\begin{align*} y^{2} y^{\prime \prime }-a&=0 \end{align*}
✓ Solution by Maple
Time used: 0.112 (sec). Leaf size: 369
dsolve(y(x)^2*diff(diff(y(x),x),x)-a=0,y(x), singsol=all)
\begin{align*}
y &= \frac {c_{1} \left (a^{2} c_{1}^{2}+2 a c_{1} {\mathrm e}^{\operatorname {RootOf}\left (\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) c_{1}^{4} a^{2}-2 \textit {\_Z} \,c_{1}^{3} a \,{\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^{2}-2 \textit {\_Z} \,c_{1}^{3} a \,{\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^{2}-2 \textit {\_Z} \,c_{1}^{3} a \,{\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 &= \frac {c_{1} \left (a^{2} c_{1}^{2}+2 a c_{1} {\mathrm e}^{\operatorname {RootOf}\left (\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) c_{1}^{4} a^{2}-2 \textit {\_Z} \,c_{1}^{3} a \,{\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^{2}-2 \textit {\_Z} \,c_{1}^{3} a \,{\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^{2}-2 \textit {\_Z} \,c_{1}^{3} a \,{\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: 0.175 (sec). Leaf size: 65
DSolve[-a + y[x]^2*D[y[x],{x,2}] == 0,y[x],x,IncludeSingularSolutions -> True]
\[
\text {Solve}\left [\left (\frac {2 a \text {arctanh}\left (\frac {\sqrt {-\frac {2 a}{y(x)}+c_1}}{\sqrt {c_1}}\right )}{c_1{}^{3/2}}+\frac {y(x) \sqrt {-\frac {2 a}{y(x)}+c_1}}{c_1}\right ){}^2=(x+c_2){}^2,y(x)\right ]
\]