7.230 problem 1821 (book 6.230)

Internal problem ID [10142]
Internal file name [OUTPUT/9089_Monday_June_06_2022_06_28_40_AM_54557410/index.tex]

Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1821 (book 6.230).
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.

\[ \boxed {\left (\operatorname {f1} y^{\prime }+\operatorname {f2} y\right ) y^{\prime \prime }+\operatorname {f3} {y^{\prime }}^{2}+\operatorname {f4} \left (x \right ) y y^{\prime }+\operatorname {f5} \left (x \right ) y^{2}=0} \]

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 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 [0, y] 
-> Calling odsolve with the ODE`, diff(_b(_a), _a) = -(_b(_a)^3*f1+_b(_a)^2*f2+_b(_a)^2*f3+f4(_a)*_b(_a)+f5(_a))/(_b(_a)*f1+f2), _b( 
   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`[0, y]
 

✗ Solution by Maple

dsolve((f1*diff(y(x),x)+f2*y(x))*diff(diff(y(x),x),x)+f3*diff(y(x),x)^2+f4(x)*y(x)*diff(y(x),x)+f5(x)*y(x)^2=0,y(x), singsol=all)
 

\[ \text {No solution found} \]

✗ Solution by Mathematica

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

DSolve[f5[x]*y[x]^2 + f4[x]*y[x]*y'[x] + f3[x]*y'[x]^2 + (f2[x]*y[x] + f1[x]*y'[x])*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
 

Timed out