Internal problem ID [10105]
Internal file name [OUTPUT/9052_Monday_June_06_2022_06_17_40_AM_70828900/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1784 (book 6.193).
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_y_y1]]
Unable to solve or complete the solution.
\[ \boxed {\left (x +y^{2}\right ) y^{\prime \prime }-2 \left (x -y^{2}\right ) {y^{\prime }}^{3}+y^{\prime } \left (1+4 y y^{\prime }\right )=0} \]
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 a change of variables {x -> y(x), y(x) -> x} 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 differential order: 2; missing variables -> 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 a change of variables {x -> y(x), y(x) -> x} and re-entering methods for dynamical symmetries --- -> trying 2nd order, dynamical_symmetries, fully reducible to Abel through one integrating factor of the form G(x,y)/(1+H(x,y)*y) trying 2nd order, integrating factors of the form mu(x,y)/(y)^n, only the singular cases *** Sublevel 2 *** Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear <- 1st order linear successful -> Calling odsolve with the ODE`, diff(diff(y(x), x), x), y(x)` *** Sublevel 2 *** Methods for second order ODEs: --- Trying classification methods --- trying a quadrature <- quadrature successful integrating factor(s) found: 1/(x+y(x)^2)/diff(y(x),x)^2, y(x)/(x+y(x)^2)/diff(y(x),x)^2`
✓ Solution by Maple
Time used: 0.14 (sec). Leaf size: 41
dsolve((x+y(x)^2)*diff(diff(y(x),x),x)-2*(x-y(x)^2)*diff(y(x),x)^3+diff(y(x),x)*(1+4*y(x)*diff(y(x),x))=0,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= \sqrt {-x} \\ y \left (x \right ) &= -\sqrt {-x} \\ \frac {-c_{1} y \left (x \right )+\ln \left (x +y \left (x \right )^{2}\right )+c_{2} +2}{y \left (x \right )} &= 0 \\ \end{align*}
✓ Solution by Mathematica
Time used: 1.242 (sec). Leaf size: 26
DSolve[-2*(x - y[x]^2)*y'[x]^3 + y'[x]*(1 + 4*y[x]*y'[x]) + (x + y[x]^2)*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [x=-y(x)^2+c_2 e^{e^{-c_1} y(x)},y(x)\right ] \]