Internal problem ID [10140]
Internal file name [OUTPUT/9087_Monday_June_06_2022_06_28_15_AM_61941447/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1819 (book 6.228).
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.
\[ \boxed {\left (y^{\prime } x -y\right ) y^{\prime \prime }-\left ({y^{\prime }}^{2}+1\right )^{2}=0} \]
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: 2 linear symmetries. Trying reduction of order `, `2nd order, trying reduction of order with given symmetries:`[x, y], [y, -x]
✓ Solution by Maple
Time used: 0.109 (sec). Leaf size: 80
dsolve((x*diff(y(x),x)-y(x))*diff(diff(y(x),x),x)-(diff(y(x),x)^2+1)^2=0,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= -i x \\ y \left (x \right ) &= i x \\ y \left (x \right ) &= \operatorname {RootOf}\left (-\ln \left (x \right )+\int _{}^{\textit {\_Z}}\frac {-\textit {\_f} +\operatorname {RootOf}\left (-c_{1} \tan \left (\frac {1}{\textit {\_Z}}\right ) \textit {\_Z} +\textit {\_f} c_{1} \tan \left (\frac {1}{\textit {\_Z}}\right )+c_{1} \textit {\_Z} \textit {\_f} +\tan \left (\frac {1}{\textit {\_Z}}\right ) \textit {\_Z} \textit {\_f} +c_{1} +\tan \left (\frac {1}{\textit {\_Z}}\right )+\textit {\_Z} -\textit {\_f} \right )}{\textit {\_f}^{2}+1}d \textit {\_f} +c_{2} \right ) x \\ \end{align*}
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[-(1 + y'[x]^2)^2 + (-y[x] + x*y'[x])*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
Not solved