Internal problem ID [10129]
Internal file name [OUTPUT/9076_Monday_June_06_2022_06_22_31_AM_71701637/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1808 (book 6.217).
ODE order: 2.
ODE degree: 1.
The type(s) of ODE detected by this program : "unknown"
Maple gives the following as the ode type
[[_Painleve, `6th`]]
Unable to solve or complete the solution.
\[ \boxed {2 x^{2} y \left (1-x \right )^{2} \left (1-y\right ) \left (x -y\right ) y^{\prime \prime }-x^{2} \left (1-x \right )^{2} \left (x -2 y x -2 y+3 y^{2}\right ) {y^{\prime }}^{2}-2 x y \left (1-x \right ) \left (1-y\right ) \left (x^{2}+y-2 y x \right ) y^{\prime }+b x \left (1-y\right )^{2} \left (x -y\right )^{2}-c \left (1-x \right ) y^{2} \left (x -y\right )^{2}-d x y^{2} \left (1-x \right ) \left (1-y\right )^{2}+a y^{2} \left (x -y\right )^{2} \left (1-y\right )^{2}=0} \]
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 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`
✗ Solution by Maple
dsolve(2*x^2*y(x)*(1-x)^2*(1-y(x))*(x-y(x))*diff(diff(y(x),x),x)-x^2*(1-x)^2*(x-2*x*y(x)-2*y(x)+3*y(x)^2)*diff(y(x),x)^2-2*x*y(x)*(1-x)*(1-y(x))*(x^2+y(x)-2*x*y(x))*diff(y(x),x)+b*x*(1-y(x))^2*(x-y(x))^2-c*(1-x)*y(x)^2*(x-y(x))^2-d*x*y(x)^2*(1-x)*(1-y(x))^2+a*y(x)^2*(x-y(x))^2*(1-y(x))^2=0,y(x), singsol=all)
\[ \text {No solution found} \]
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[b*x*(1 - y[x])^2*(x - y[x])^2 - d*(1 - x)*x*(1 - y[x])^2*y[x]^2 - c*(1 - x)*(x - y[x])^2*y[x]^2 + a*(1 - y[x])^2*(x - y[x])^2*y[x]^2 - 2*(1 - x)*x*(1 - y[x])*y[x]*(x^2 + y[x] - 2*x*y[x])*y'[x] - (1 - x)^2*x^2*(x - 2*y[x] - 2*x*y[x] + 3*y[x]^2)*y'[x]^2 + 2*(1 - x)^2*x^2*(1 - y[x])*(x - y[x])*y[x]*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
Not solved