Internal problem ID [10117]
Internal file name [OUTPUT/9064_Monday_June_06_2022_06_19_32_AM_96042846/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1796 (book 6.205).
ODE order: 2.
ODE degree: 1.
The type(s) of ODE detected by this program : "unknown"
Maple gives the following as the ode type
[[_Emden, _Fowler], [_2nd_order, _with_linear_symmetries]]
Unable to solve or complete the solution.
\[ \boxed {x y^{2} y^{\prime \prime }=a} \]
Maple trace
`Methods for second order ODEs: --- Trying classification methods --- trying 2nd order Liouville trying 2nd order WeierstrassP trying 2nd order JacobiSN differential order: 2; trying a linearization to 3rd order trying 2nd order ODE linearizable_by_differentiation trying 2nd order, 2 integrating factors of the form mu(x,y) trying differential order: 2; missing variables -> trying 2nd order, dynamical_symmetries, fully reducible to Abel through one integrating factor of the form G(x,y)/(1+H(x,y)*y)^2 trying 2nd order, integrating factors of the form mu(x,y)/(y)^n, only the singular cases trying symmetries linear in x and y(x) trying differential order: 2; exact nonlinear trying 2nd order, integrating factor of the form mu(y) trying 2nd order, integrating factor of the form mu(x,y) trying 2nd order, integrating factor of the form mu(x,y)/(y)^n, only the general case trying 2nd order, integrating factor of the form mu(y,y) trying differential order: 2; mu polynomial in y trying 2nd order, integrating factor of the form mu(x,y) differential order: 2; looking for linear symmetries differential order: 2; found: 1 linear symmetries. Trying reduction of order `, `2nd order, trying reduction of order with given symmetries:`[3*x, y]
✓ Solution by Maple
Time used: 0.078 (sec). Leaf size: 793
dsolve(x*y(x)^2*diff(diff(y(x),x),x)-a=0,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= \frac {c_{1} x \left (81 c_{1}^{2} a^{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 \,{\mathrm e}^{2 \textit {\_Z}} \operatorname {csgn}\left (\frac {1}{c_{1}}\right ) c_{1}^{2} x -6 \,{\mathrm e}^{\textit {\_Z}} \operatorname {csgn}\left (\frac {1}{c_{1}}\right ) c_{2} x -2 \,{\mathrm e}^{\textit {\_Z}} \operatorname {csgn}\left (\frac {1}{c_{1}}\right )\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 \,{\mathrm e}^{2 \textit {\_Z}} \operatorname {csgn}\left (\frac {1}{c_{1}}\right ) c_{1}^{2} x -6 \,{\mathrm e}^{\textit {\_Z}} \operatorname {csgn}\left (\frac {1}{c_{1}}\right ) c_{2} x -2 \,{\mathrm e}^{\textit {\_Z}} \operatorname {csgn}\left (\frac {1}{c_{1}}\right )\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 \,{\mathrm e}^{2 \textit {\_Z}} \operatorname {csgn}\left (\frac {1}{c_{1}}\right ) c_{1}^{2} x -6 \,{\mathrm e}^{\textit {\_Z}} \operatorname {csgn}\left (\frac {1}{c_{1}}\right ) c_{2} x -2 \,{\mathrm e}^{\textit {\_Z}} \operatorname {csgn}\left (\frac {1}{c_{1}}\right )\right )}}{2} \\ y \left (x \right ) &= \frac {c_{1} x \left (81 c_{1}^{2} a^{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 \,{\mathrm e}^{2 \textit {\_Z}} \operatorname {csgn}\left (\frac {1}{c_{1}}\right ) c_{1}^{2} x +6 \,{\mathrm e}^{\textit {\_Z}} \operatorname {csgn}\left (\frac {1}{c_{1}}\right ) c_{2} x +2 \,{\mathrm e}^{\textit {\_Z}} \operatorname {csgn}\left (\frac {1}{c_{1}}\right )\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 \,{\mathrm e}^{2 \textit {\_Z}} \operatorname {csgn}\left (\frac {1}{c_{1}}\right ) c_{1}^{2} x +6 \,{\mathrm e}^{\textit {\_Z}} \operatorname {csgn}\left (\frac {1}{c_{1}}\right ) c_{2} x +2 \,{\mathrm e}^{\textit {\_Z}} \operatorname {csgn}\left (\frac {1}{c_{1}}\right )\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 \,{\mathrm e}^{2 \textit {\_Z}} \operatorname {csgn}\left (\frac {1}{c_{1}}\right ) c_{1}^{2} x +6 \,{\mathrm e}^{\textit {\_Z}} \operatorname {csgn}\left (\frac {1}{c_{1}}\right ) c_{2} x +2 \,{\mathrm e}^{\textit {\_Z}} \operatorname {csgn}\left (\frac {1}{c_{1}}\right )\right )}}{2} \\ y \left (x \right ) &= \frac {c_{1} x \left (81 c_{1}^{2} a^{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 \,{\mathrm e}^{2 \textit {\_Z}} \operatorname {csgn}\left (\frac {1}{c_{1}}\right ) c_{1}^{2} x +6 \,{\mathrm e}^{\textit {\_Z}} \operatorname {csgn}\left (\frac {1}{c_{1}}\right ) c_{2} x +2 \,{\mathrm e}^{\textit {\_Z}} \operatorname {csgn}\left (\frac {1}{c_{1}}\right )\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 \,{\mathrm e}^{2 \textit {\_Z}} \operatorname {csgn}\left (\frac {1}{c_{1}}\right ) c_{1}^{2} x +6 \,{\mathrm e}^{\textit {\_Z}} \operatorname {csgn}\left (\frac {1}{c_{1}}\right ) c_{2} x +2 \,{\mathrm e}^{\textit {\_Z}} \operatorname {csgn}\left (\frac {1}{c_{1}}\right )\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 \,{\mathrm e}^{2 \textit {\_Z}} \operatorname {csgn}\left (\frac {1}{c_{1}}\right ) c_{1}^{2} x +6 \,{\mathrm e}^{\textit {\_Z}} \operatorname {csgn}\left (\frac {1}{c_{1}}\right ) c_{2} x +2 \,{\mathrm e}^{\textit {\_Z}} \operatorname {csgn}\left (\frac {1}{c_{1}}\right )\right )}}{2} \\ y \left (x \right ) &= \frac {c_{1} x \left (81 c_{1}^{2} a^{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 \,{\mathrm e}^{2 \textit {\_Z}} \operatorname {csgn}\left (\frac {1}{c_{1}}\right ) c_{1}^{2} x -6 \,{\mathrm e}^{\textit {\_Z}} \operatorname {csgn}\left (\frac {1}{c_{1}}\right ) c_{2} x -2 \,{\mathrm e}^{\textit {\_Z}} \operatorname {csgn}\left (\frac {1}{c_{1}}\right )\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 \,{\mathrm e}^{2 \textit {\_Z}} \operatorname {csgn}\left (\frac {1}{c_{1}}\right ) c_{1}^{2} x -6 \,{\mathrm e}^{\textit {\_Z}} \operatorname {csgn}\left (\frac {1}{c_{1}}\right ) c_{2} x -2 \,{\mathrm e}^{\textit {\_Z}} \operatorname {csgn}\left (\frac {1}{c_{1}}\right )\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 \,{\mathrm e}^{2 \textit {\_Z}} \operatorname {csgn}\left (\frac {1}{c_{1}}\right ) c_{1}^{2} x -6 \,{\mathrm e}^{\textit {\_Z}} \operatorname {csgn}\left (\frac {1}{c_{1}}\right ) c_{2} x -2 \,{\mathrm e}^{\textit {\_Z}} \operatorname {csgn}\left (\frac {1}{c_{1}}\right )\right )}}{2} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.305 (sec). Leaf size: 116
DSolve[-a + x*y[x]^2*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [-\frac {a \arctan \left (\frac {\sqrt {2} \sqrt {c_1} \left (\frac {y(x)}{x}+\frac {a}{2 c_1}\right )}{\sqrt {-\frac {2 a y(x)}{x}-\frac {2 c_1 y(x)^2}{x^2}}}\right )}{2 \sqrt {2} c_1{}^{3/2}}-\frac {\sqrt {-\frac {2 a y(x)}{x}-\frac {2 c_1 y(x)^2}{x^2}}}{2 c_1}-\frac {1}{x}-c_2=0,y(x)\right ] \]