7.174 problem 1765 (book 6.174)

Internal problem ID [10086]
Internal file name [OUTPUT/9033_Monday_June_06_2022_06_15_30_AM_83579840/index.tex]

Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1765 (book 6.174).
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_x_y1], [_2nd_order, _reducible, _mu_xy]]

Unable to solve or complete the solution.

\[ \boxed {x y y^{\prime \prime }-2 x {y^{\prime }}^{2}+\left (y+1\right ) y^{\prime }=0} \]

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 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 
trying symmetries linear in x and y(x) 
`, `-> Computing symmetries using: way = 3 
Try integration with the canonical coordinates of the symmetry [x, 0] 
-> Calling odsolve with the ODE`, diff(_b(_a), _a) = _b(_a)*(_b(_a)-2)/_a, _b(_a), explicit, HINT = [[_a, 0]]`   *** Sublevel 2 *** 
   symmetry methods on request 
`, `1st order, trying reduction of order with given symmetries:`[_a, 0]
 

Solution by Maple

Time used: 0.094 (sec). Leaf size: 22

dsolve(x*y(x)*diff(diff(y(x),x),x)-2*x*diff(y(x),x)^2+(y(x)+1)*diff(y(x),x)=0,y(x), singsol=all)
 

\begin{align*} y \left (x \right ) &= 0 \\ y \left (x \right ) &= c_{1} \tanh \left (\frac {\ln \left (x \right )-c_{2}}{2 c_{1}}\right ) \\ \end{align*}

Solution by Mathematica

Time used: 34.063 (sec). Leaf size: 52

DSolve[(1 + y[x])*y'[x] - 2*x*y'[x]^2 + x*y[x]*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \frac {\tan \left (\frac {\sqrt {c_1} (\log (x)-c_2)}{\sqrt {2}}\right )}{\sqrt {2} \sqrt {c_1}} \\ y(x)\to \frac {1}{2} (\log (x)-c_2) \\ \end{align*}