Internal problem ID [10097]
Internal file name [OUTPUT/9044_Monday_June_06_2022_06_16_38_AM_36426526/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1776 (book 6.185).
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.
\[ \boxed {x \left (x +1\right )^{2} y y^{\prime \prime }-x \left (x +1\right )^{2} {y^{\prime }}^{2}+2 \left (x +1\right )^{2} y y^{\prime }-a \left (x +2\right ) y^{2}=0} \]
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: 0.062 (sec). Leaf size: 31
dsolve(x*(x+1)^2*y(x)*diff(diff(y(x),x),x)-x*(x+1)^2*diff(y(x),x)^2+2*(x+1)^2*y(x)*diff(y(x),x)-a*(x+2)*y(x)^2=0,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= 0 \\ y \left (x \right ) &= \frac {\left (1+x \right )^{a} {\mathrm e}^{\frac {\left (-1-x \right ) a +c_{1}}{x}}}{c_{2}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.661 (sec). Leaf size: 24
DSolve[-(a*(2 + x)*y[x]^2) + 2*(1 + x)^2*y[x]*y'[x] - x*(1 + x)^2*y'[x]^2 + x*(1 + x)^2*y[x]*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to c_2 (x+1)^a e^{-\frac {a+c_1}{x}} \]