Internal problem ID [9982]
Internal file name [OUTPUT/8929_Monday_June_06_2022_05_59_05_AM_4771993/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1661 (book 6.70).
ODE order: 2.
ODE degree: 0.
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.
\[ \boxed {y^{\prime \prime }-x^{-2+n} f \left (y x^{-n}, y^{\prime } x^{1-n}\right )=0} \]
Maple trace
`Methods for second order ODEs: --- Trying classification methods --- 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 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: 1 linear symmetries. Trying reduction of order `, `2nd order, trying reduction of order with given symmetries:`[x, n*y]
✗ Solution by Maple
dsolve(diff(diff(y(x),x),x)-x^(n-2)*f(y(x)/(x^n),diff(y(x),x)/(x^(n-1)))=0,y(x), singsol=all)
\[ \text {No solution found} \]
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[-(x^(-2 + n)*f[y[x]/x^n, x^(1 - n)*y'[x]]) + y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
Not solved