Internal problem ID [10139]
Internal file name [OUTPUT/9086_Monday_June_06_2022_06_28_09_AM_10929050/index.tex
]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1818 (book 6.227).
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.
\[ \boxed {\left (y^{\prime } x -y\right ) y^{\prime \prime }+4 {y^{\prime }}^{2}=0} \]
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) <- linear symmetries successful`
✓ Solution by Maple
Time used: 0.063 (sec). Leaf size: 44
dsolve((x*diff(y(x),x)-y(x))*diff(diff(y(x),x),x)+4*diff(y(x),x)^2=0,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= 0 \\ y \left (x \right ) &= {\mathrm e}^{\int _{}^{\ln \left (x \right )}\left ({\mathrm e}^{\operatorname {RootOf}\left (\ln \left ({\mathrm e}^{\textit {\_Z}}-1\right ) {\mathrm e}^{\textit {\_Z}}+c_{1} {\mathrm e}^{\textit {\_Z}}-\textit {\_Z} \,{\mathrm e}^{\textit {\_Z}}-\textit {\_b} \,{\mathrm e}^{\textit {\_Z}}+2\right )}-1\right )d \textit {\_b} +c_{2}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 75.536 (sec). Leaf size: 41
DSolve[4*y'[x]^2 + (-y[x] + x*y'[x])*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {1}{2} c_2 e^{-2-W\left (\frac {2 x}{e^2 c_1}\right )} \left (2+W\left (\frac {2 x}{e^2 c_1}\right )\right ) \]