Internal problem ID [7108]
Internal file name [OUTPUT/6094_Sunday_June_05_2022_04_21_36_PM_24970600/index.tex]
Book: Own collection of miscellaneous problems
Section: section 1.0
Problem number: 64.
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.
\[ \boxed {y^{2} y^{\prime \prime }=x} \]
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) -> Calling odsolve with the ODE`, -(_y1^3-4)*y(x)/_y1^3+2*((diff(y(x), x))*x+_y1)/_y1^3, y(x)` *** Sublevel 2 *** Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear <- 1st order linear successful 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:`[x, y]
✓ Solution by Maple
Time used: 0.078 (sec). Leaf size: 106
dsolve(y(x)^2*diff(y(x),x$2)=x,y(x), singsol=all)
\[ y \left (x \right ) = \operatorname {RootOf}\left (\ln \left (x \right )+2^{\frac {1}{3}} \left (\int _{}^{\textit {\_Z}}\frac {1}{2^{\frac {1}{3}} \textit {\_f} +2 \operatorname {RootOf}\left (\operatorname {AiryBi}\left (\frac {2 \textit {\_Z}^{2} \textit {\_f} +2^{\frac {2}{3}}}{2 \textit {\_f}}\right ) c_{1} \textit {\_Z} +\textit {\_Z} \operatorname {AiryAi}\left (\frac {2 \textit {\_Z}^{2} \textit {\_f} +2^{\frac {2}{3}}}{2 \textit {\_f}}\right )+\operatorname {AiryBi}\left (1, \frac {2 \textit {\_Z}^{2} \textit {\_f} +2^{\frac {2}{3}}}{2 \textit {\_f}}\right ) c_{1} +\operatorname {AiryAi}\left (1, \frac {2 \textit {\_Z}^{2} \textit {\_f} +2^{\frac {2}{3}}}{2 \textit {\_f}}\right )\right )}d \textit {\_f} \right )-c_{2} \right ) x \]
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[y[x]^2*y''[x]==x,y[x],x,IncludeSingularSolutions -> True]
Not solved