problem 1739 (book 6.148)

Internal problem ID [10060]
Internal ﬁle name [OUTPUT/9007_Monday_June_06_2022_06_11_25_AM_52502799/index.tex]

Book: Diﬀerential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1739 (book 6.148).
ODE order: 2.
ODE degree: 1.

\[ \boxed {2 y^{\prime \prime } y-{y^{\prime }}^{2}+3 f \left (x \right ) y^{\prime } y+2 \left (f \left (x \right )^{2}+f^{\prime }\left (x \right )\right ) y^{2}-8 y^{3}=0} \]

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) 
trying differential order: 2; exact nonlinear 
trying 2nd order, integrating factor of the form mu(y) 
trying 2nd order, integrating factor of the form mu(x,y) 
trying 2nd order, integrating factor of the form mu(x,y)/(y)^n, only the general case 
trying 2nd order, integrating factor of the form mu(y,y) 
trying differential order: 2; mu polynomial in y 
trying 2nd order, integrating factor of the form mu(x,y) 
differential order: 2; looking for linear symmetries 
-> trying 2nd order, the S-function method 
   -> trying a change of variables {x -> y(x), y(x) -> x} and re-entering methods for the S-function 
   -> trying 2nd order, the S-function method 
   -> trying 2nd order, No Point Symmetries Class V 
      --- trying a change of variables {x -> y(x), y(x) -> x} and re-entering methods for dynamical symmetries --- 
      -> trying 2nd order, No Point Symmetries Class V 
   -> trying 2nd order, No Point Symmetries Class V 
      --- trying a change of variables {x -> y(x), y(x) -> x} and re-entering methods for dynamical symmetries --- 
      -> trying 2nd order, No Point Symmetries Class V 
   -> trying 2nd order, No Point Symmetries Class V 
      --- trying a change of variables {x -> y(x), y(x) -> x} and re-entering methods for dynamical symmetries --- 
      -> trying 2nd order, No Point Symmetries Class V 
trying 2nd order, integrating factor of the form mu(x,y)/(y)^n, only the general case 
-> trying 2nd order, dynamical_symmetries, only a reduction of order through one integrating factor of the form G(x,y)/(1+H(x,y)*y)^ 
   --- trying a change of variables {x -> y(x), y(x) -> x} and re-entering methods for dynamical symmetries --- 
   -> trying 2nd order, dynamical_symmetries, only a reduction of order through one integrating factor of the form G(x,y)/(1+H(x,y)* 
--- Trying Lie symmetry methods, 2nd order --- 
`, `-> Computing symmetries using: way = 3 
`, `-> Computing symmetries using: way = 5 
`, `-> Computing symmetries using: way = formal 
            *** 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)-(1/2)*(diff(y(x), x))*f(x)-(1/2)*(diff(f(x), x))*y(x), y(x)`      *** Su 
      Methods for second order ODEs: 
      --- Trying classification methods --- 
      trying a symmetry of the form [xi=0, eta=F(x)]

7.148 problem 1739 (book 6.148)