Internal problem ID [9071]
Internal file name [OUTPUT/8006_Monday_June_06_2022_01_14_52_AM_10298365/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 737.
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program : "unknown"
Maple gives the following as the ode type
[_rational, [_1st_order, `_with_symmetry_[F(x),G(x)]`], [_Abel, `2nd type`, `class A`]]
Unable to solve or complete the solution.
\[ \boxed {y^{\prime }-\frac {x \left (-1+x -2 y x +2 x^{3}\right )}{x^{2}-y}=0} \] Unable to determine ODE type.
Maple trace
`Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear trying Bernoulli trying separable trying inverse linear trying homogeneous types: trying Chini differential order: 1; looking for linear symmetries trying exact trying Abel <- Abel successful`
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 28
dsolve(diff(y(x),x) = 1/(x^2-y(x))*x*(-1+x-2*x*y(x)+2*x^3),y(x), singsol=all)
\[ y \left (x \right ) = x^{2}+\frac {\operatorname {LambertW}\left (-2 c_{1} {\mathrm e}^{\frac {4}{3} x^{3}-2 x^{2}-1}\right )}{2}+\frac {1}{2} \]
✓ Solution by Mathematica
Time used: 3.498 (sec). Leaf size: 47
DSolve[y'[x] == (x*(-1 + x + 2*x^3 - 2*x*y[x]))/(x^2 - y[x]),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to x^2+\frac {1}{2} \left (1+W\left (-e^{\frac {4 x^3}{3}-2 x^2-1+c_1}\right )\right ) \\ y(x)\to x^2+\frac {1}{2} \\ \end{align*}