Internal problem ID [10101]
Internal file name [OUTPUT/9048_Monday_June_06_2022_06_17_13_AM_77686969/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1780 (book 6.189).
ODE order: 2.
ODE degree: 1.
The type(s) of ODE detected by this program : "unknown"
Maple gives the following as the ode type
[[_2nd_order, _with_linear_symmetries]]
Unable to solve or complete the solution.
\[ \boxed {y^{2} y^{\prime \prime }+{y^{\prime }}^{2} y=-a x} \]
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 a change of variables {x -> y(x), y(x) -> x} and re-entering methods for dynamical symmetries --- -> trying 2nd order, dynamical_symmetries, fully reducible to Abel through one integrating factor of the form G(x,y)/(1+H(x,y)*y) 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:`[x, y]
✓ Solution by Maple
Time used: 0.078 (sec). Leaf size: 110
dsolve(y(x)^2*diff(diff(y(x),x),x)+y(x)*diff(y(x),x)^2+a*x=0,y(x), singsol=all)
\[ \ln \left (x \right )-\frac {\left (\int _{}^{\frac {y \left (x \right )}{x}}\frac {\textit {\_g}^{2} \left (\left (\left (\frac {a}{\textit {\_g}^{3}}\right )^{\frac {1}{3}}-2\right ) \sqrt {3}+3 \left (\frac {a}{\textit {\_g}^{3}}\right )^{\frac {1}{3}} \tan \left (\operatorname {RootOf}\left (2 \textit {\_Z} \sqrt {3}-\ln \left (\frac {1}{\sqrt {3}\, \sin \left (2 \textit {\_Z} \right )+2+\cos \left (2 \textit {\_Z} \right )}\right )-6 c_{1} -6 \left (\int \frac {\left (\frac {a}{\textit {\_g}^{3}}\right )^{\frac {2}{3}} \textit {\_g}^{2}}{\textit {\_g}^{3}+a}d \textit {\_g} \right )\right )\right )\right )}{\textit {\_g}^{3}+a}d \textit {\_g} \right ) \sqrt {3}}{6}-c_{2} = 0 \]
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[a*x + y[x]*y'[x]^2 + y[x]^2*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
Not solved