Internal problem ID [10093]
Internal file name [OUTPUT/9040_Monday_June_06_2022_06_16_12_AM_16276387/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1772 (book 6.181).
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], [_2nd_order, _reducible, _mu_xy]]
Unable to solve or complete the solution.
\[ \boxed {x^{2} \left (x +y\right ) y^{\prime \prime }-\left (y^{\prime } x -y\right )^{2}=0} \]
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 a quadrature checking if the LODE has constant coefficients checking if the LODE is of Euler type <- LODE of Euler type successful <- 2nd order, 2 integrating factors of the form mu(x,y) successful`
✓ Solution by Maple
Time used: 0.047 (sec). Leaf size: 29
dsolve(x^2*(x+y(x))*diff(diff(y(x),x),x)-(x*diff(y(x),x)-y(x))^2=0,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= -x \\ y \left (x \right ) &= \frac {x \left ({\mathrm e}^{\frac {c_{1} -x}{x}}-c_{2} \right )}{c_{2}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 1.012 (sec). Leaf size: 20
DSolve[-(-y[x] + x*y'[x])^2 + x^2*(x + y[x])*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to x \left (-1+c_2 e^{\frac {c_1}{x}}\right ) \]