2.3.24 problem 24

Internal problem ID [8558]
Book : Own collection of miscellaneous problems
Section : section 3.0
Problem number : 24
Date solved : Thursday, December 12, 2024 at 09:30:54 AM
CAS classification : [NONE]

Solve

\begin{align*} \left (x^{2}+1\right ) y^{\prime \prime }+y {y^{\prime }}^{2}&=0 \end{align*}

Maple step by step solution

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) 
-> Calling odsolve with the ODE`, -(_y1^2*x^2+_y1^2-4*x^2)*y(x)/((x^2+1)*_y1^2)+(2*x^3*(diff(y(x), x))+_y1*x^2+2*(diff(y(x), x))*x-_ 
   Methods for first order ODEs: 
   --- Trying classification methods --- 
   trying a quadrature 
   trying 1st order linear 
   <- 1st order linear successful 
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`
 

Maple dsolve solution

Solving time : 0.172 (sec)
Leaf size : maple_leaf_size

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

\[ \text {No solution found} \]

Mathematica DSolve solution

Solving time : 0.0 (sec)
Leaf size : 0

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

Not solved