60.7.205 problem 1796 (book 6.205)
Internal
problem
ID
[11794]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
6,
non-linear
second
order
Problem
number
:
1796
(book
6.205)
Date
solved
:
Tuesday, January 28, 2025 at 06:11:25 PM
CAS
classification
:
[[_Emden, _Fowler], [_2nd_order, _with_linear_symmetries]]
\begin{align*} x y^{2} y^{\prime \prime }-a&=0 \end{align*}
✓ Solution by Maple
Time used: 0.125 (sec). Leaf size: 793
dsolve(x*y(x)^2*diff(diff(y(x),x),x)-a=0,y(x), singsol=all)
\begin{align*}
y &= \frac {x c_{1} \left (81 a^{2} c_{1}^{2}+18 a c_{1} {\mathrm e}^{\operatorname {RootOf}\left (243 \,\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) c_{1}^{4} a^{2} x -54 \textit {\_Z} \,{\mathrm e}^{\textit {\_Z}} a x \,c_{1}^{3}-3 \,\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{2 \textit {\_Z}} c_{1}^{2} x -6 \,\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{\textit {\_Z}} c_{2} x -2 \,\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{\textit {\_Z}}\right )}+{\mathrm e}^{2 \operatorname {RootOf}\left (243 \,\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) c_{1}^{4} a^{2} x -54 \textit {\_Z} \,{\mathrm e}^{\textit {\_Z}} a x \,c_{1}^{3}-3 \,\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{2 \textit {\_Z}} c_{1}^{2} x -6 \,\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{\textit {\_Z}} c_{2} x -2 \,\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{\textit {\_Z}}\right )}\right ) {\mathrm e}^{-\operatorname {RootOf}\left (243 \,\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) c_{1}^{4} a^{2} x -54 \textit {\_Z} \,{\mathrm e}^{\textit {\_Z}} a x \,c_{1}^{3}-3 \,\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{2 \textit {\_Z}} c_{1}^{2} x -6 \,\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{\textit {\_Z}} c_{2} x -2 \,\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{\textit {\_Z}}\right )}}{2} \\
y &= \frac {x c_{1} \left (81 a^{2} c_{1}^{2}+18 a c_{1} {\mathrm e}^{\operatorname {RootOf}\left (243 \,\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) c_{1}^{4} a^{2} x +54 \textit {\_Z} \,{\mathrm e}^{\textit {\_Z}} a x \,c_{1}^{3}-3 \,\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{2 \textit {\_Z}} c_{1}^{2} x +6 \,\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{\textit {\_Z}} c_{2} x +2 \,\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{\textit {\_Z}}\right )}+{\mathrm e}^{2 \operatorname {RootOf}\left (243 \,\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) c_{1}^{4} a^{2} x +54 \textit {\_Z} \,{\mathrm e}^{\textit {\_Z}} a x \,c_{1}^{3}-3 \,\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{2 \textit {\_Z}} c_{1}^{2} x +6 \,\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{\textit {\_Z}} c_{2} x +2 \,\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{\textit {\_Z}}\right )}\right ) {\mathrm e}^{-\operatorname {RootOf}\left (243 \,\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) c_{1}^{4} a^{2} x +54 \textit {\_Z} \,{\mathrm e}^{\textit {\_Z}} a x \,c_{1}^{3}-3 \,\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{2 \textit {\_Z}} c_{1}^{2} x +6 \,\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{\textit {\_Z}} c_{2} x +2 \,\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{\textit {\_Z}}\right )}}{2} \\
y &= \frac {x c_{1} \left (81 a^{2} c_{1}^{2}+18 a c_{1} {\mathrm e}^{\operatorname {RootOf}\left (243 \,\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) c_{1}^{4} a^{2} x -54 \textit {\_Z} \,{\mathrm e}^{\textit {\_Z}} a x \,c_{1}^{3}-3 \,\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{2 \textit {\_Z}} c_{1}^{2} x +6 \,\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{\textit {\_Z}} c_{2} x +2 \,\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{\textit {\_Z}}\right )}+{\mathrm e}^{2 \operatorname {RootOf}\left (243 \,\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) c_{1}^{4} a^{2} x -54 \textit {\_Z} \,{\mathrm e}^{\textit {\_Z}} a x \,c_{1}^{3}-3 \,\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{2 \textit {\_Z}} c_{1}^{2} x +6 \,\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{\textit {\_Z}} c_{2} x +2 \,\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{\textit {\_Z}}\right )}\right ) {\mathrm e}^{-\operatorname {RootOf}\left (243 \,\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) c_{1}^{4} a^{2} x -54 \textit {\_Z} \,{\mathrm e}^{\textit {\_Z}} a x \,c_{1}^{3}-3 \,\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{2 \textit {\_Z}} c_{1}^{2} x +6 \,\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{\textit {\_Z}} c_{2} x +2 \,\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{\textit {\_Z}}\right )}}{2} \\
y &= \frac {x c_{1} \left (81 a^{2} c_{1}^{2}+18 a c_{1} {\mathrm e}^{\operatorname {RootOf}\left (243 \,\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) c_{1}^{4} a^{2} x +54 \textit {\_Z} \,{\mathrm e}^{\textit {\_Z}} a x \,c_{1}^{3}-3 \,\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{2 \textit {\_Z}} c_{1}^{2} x -6 \,\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{\textit {\_Z}} c_{2} x -2 \,\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{\textit {\_Z}}\right )}+{\mathrm e}^{2 \operatorname {RootOf}\left (243 \,\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) c_{1}^{4} a^{2} x +54 \textit {\_Z} \,{\mathrm e}^{\textit {\_Z}} a x \,c_{1}^{3}-3 \,\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{2 \textit {\_Z}} c_{1}^{2} x -6 \,\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{\textit {\_Z}} c_{2} x -2 \,\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{\textit {\_Z}}\right )}\right ) {\mathrm e}^{-\operatorname {RootOf}\left (243 \,\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) c_{1}^{4} a^{2} x +54 \textit {\_Z} \,{\mathrm e}^{\textit {\_Z}} a x \,c_{1}^{3}-3 \,\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{2 \textit {\_Z}} c_{1}^{2} x -6 \,\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{\textit {\_Z}} c_{2} x -2 \,\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{\textit {\_Z}}\right )}}{2} \\
\end{align*}
✓ Solution by Mathematica
Time used: 0.198 (sec). Leaf size: 116
DSolve[-a + x*y[x]^2*D[y[x],{x,2}] == 0,y[x],x,IncludeSingularSolutions -> True]
\[
\text {Solve}\left [-\frac {a \arctan \left (\frac {\sqrt {2} \sqrt {c_1} \left (\frac {y(x)}{x}+\frac {a}{2 c_1}\right )}{\sqrt {-\frac {2 a y(x)}{x}-\frac {2 c_1 y(x)^2}{x^2}}}\right )}{2 \sqrt {2} c_1{}^{3/2}}-\frac {\sqrt {-\frac {2 a y(x)}{x}-\frac {2 c_1 y(x)^2}{x^2}}}{2 c_1}-\frac {1}{x}-c_2=0,y(x)\right ]
\]