7.82 problem 1673 (book 6.82)

Internal problem ID [9994]
Internal file name [OUTPUT/8941_Monday_June_06_2022_06_00_54_AM_93658782/index.tex]

Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1673 (book 6.82).
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]]

Unable to solve or complete the solution.

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

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 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 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 
`, `-> Computing symmetries using: way = exp_sym 
Try integration with the canonical coordinates of the symmetry [x, 0] 
-> Calling odsolve with the ODE`, diff(_b(_a), _a) = -_b(_a)^3*_a^n*a+_b(_a)^3*a*_a-_b(_a)^2, _b(_a), explicit`   *** Sublevel 2 *** 
   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 
   Looking for potential symmetries 
   Looking for potential symmetries 
   Looking for potential symmetries 
   trying inverse_Riccati 
   trying an equivalence to an Abel ODE 
   differential order: 1; trying a linearization to 2nd order 
   --- trying a change of variables {x -> y(x), y(x) -> x} 
   differential order: 1; trying a linearization to 2nd order 
   trying 1st order ODE linearizable_by_differentiation 
trying differential order: 2; exact nonlinear 
trying 2nd order, integrating factor of the form mu(x,y) 
-> 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 2nd order, No Point Symmetries Class V 
   -> 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 Lie symmetry methods, 2nd order --- 
`, `-> Computing symmetries using: way = 3`[x, 0]
 

✗ Solution by Maple

dsolve(x^2*diff(diff(y(x),x),x)=a*(y(x)^n-y(x)),y(x), singsol=all)
 

\[ \text {No solution found} \]

✗ Solution by Mathematica

Time used: 0.0 (sec). Leaf size: 0

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

Not solved