Internal problem ID [10110]
Internal file name [OUTPUT/9057_Monday_June_06_2022_06_18_45_AM_46697477/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1789 (book 6.198).
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, _reducible, _mu_xy]]
Unable to solve or complete the solution.
\[ \boxed {2 y \left (-1+y\right ) y^{\prime \prime }-\left (3 y-1\right ) {y^{\prime }}^{2}+4 y y^{\prime } \left (f \left (x \right ) y+g \left (x \right )\right )+4 y^{2} \left (-1+y\right ) \left (g \left (x \right )^{2}-f \left (x \right )^{2}-g^{\prime }\left (x \right )-f^{\prime }\left (x \right )\right )=0} \]
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 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 integrating factor(s) found: 1/(-1+x)/x^(1/2)*exp(Intat(f(_a)-g(_a),_a = y(x)))/diff(y(x),x)^2 attempting the computation of a related first integral... -> Calling odsolve with the ODE`, exp(Int(f(_a), _a)-(Int(g(_a), _a)))*(diff(_b(_a), _a))/((_b(_a)-1)*_b(_a)^(1/2))-(2*exp(Int(f( 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 Looking for potential symmetries trying inverse_Riccati trying an equivalence to an Abel ODE differential order: 1; trying a linearization to 2nd order -> Calling odsolve with the ODE`, diff(diff(_b(_a), _a), _a) = (1/4)*exp(-2*(Int(f(_a), _a))+2*(Int(g(_a), _a)))*c__1^2*_b(_a) Methods for second order ODEs: <- differential order: 1; linearization to 2nd order successful <- differential order: 2; exact nonlinear successful <- 2nd_order mu_xyp successful`
✗ Solution by Maple
dsolve(2*y(x)*(-1+y(x))*diff(diff(y(x),x),x)-(3*y(x)-1)*diff(y(x),x)^2+4*y(x)*diff(y(x),x)*(f(x)*y(x)+g(x))+4*y(x)^2*(-1+y(x))*(g(x)^2-f(x)^2-diff(g(x),x)-diff(f(x),x))=0,y(x), singsol=all)
\[ \text {No solution found} \]
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[-4*(1 - y[x])*y[x]^2*(-f[x]^2 + g[x]^2 - Derivative[1][f][x] - Derivative[1][g][x]) + 4*y[x]*(g[x] + f[x]*y[x])*y'[x] + (1 - 3*y[x])*y'[x]^2 - 2*(1 - y[x])*y[x]*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
Not solved