Internal problem ID [10094]
Internal file name [OUTPUT/9041_Monday_June_06_2022_06_16_17_AM_76183212/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1773 (book 6.182).
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.
\[ \boxed {x^{2} \left (x -y\right ) y^{\prime \prime }+a \left (y^{\prime } x -y\right )^{2}=0} \]
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: 45
dsolve(x^2*(x-y(x))*diff(diff(y(x),x),x)+a*(x*diff(y(x),x)-y(x))^2=0,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= x \\ y \left (x \right ) &= -\operatorname {RootOf}\left (c_{1} a x \,\textit {\_Z}^{a}-c_{1} x \,\textit {\_Z}^{a}-c_{2} a \,\textit {\_Z}^{a}+c_{2} \textit {\_Z}^{a}+\textit {\_Z} \,x^{a}\right )+x \\ \end{align*}
✓ Solution by Mathematica
Time used: 60.663 (sec). Leaf size: 36
DSolve[a*(-y[x] + x*y'[x])^2 + x^2*(x - y[x])*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to x \left (1+\left (-\frac {(a-1) \left ((-1)^a c_1+c_2 x\right )}{x}\right ){}^{\frac {1}{1-a}}\right ) \]