Internal problem ID [10005]
Internal file name [OUTPUT/8952_Monday_June_06_2022_06_02_38_AM_28761829/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1684 (book 6.93).
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.
\[ \boxed {y^{\prime \prime } x^{3}-a \left (y^{\prime } x -y\right )^{2}=0} \]
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 <- quadrature successful <- 2nd order, 2 integrating factors of the form mu(x,y) successful`
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 23
dsolve(x^3*diff(diff(y(x),x),x)-a*(x*diff(y(x),x)-y(x))^2=0,y(x), singsol=all)
\[ y \left (x \right ) = -\frac {\ln \left (\frac {a \left (c_{1} x -c_{2} \right )}{x}\right ) x}{a} \]
✓ Solution by Mathematica
Time used: 4.57 (sec). Leaf size: 25
DSolve[-(a*(-y[x] + x*y'[x])^2) + x^3*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to -\frac {x \log \left (-\frac {a (c_2 x+c_1)}{x}\right )}{a} \]