Internal problem ID [10028]
Internal file name [OUTPUT/8975_Monday_June_06_2022_06_06_35_AM_79403660/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1707 (book 6.116).
ODE order: 2.
ODE degree: 1.
The type(s) of ODE detected by this program : "unknown"
Maple gives the following as the ode type
[NONE]
Unable to solve or complete the solution.
\[ \boxed {y^{\prime \prime } y-{y^{\prime }}^{2}+f^{\prime }\left (x \right ) y^{\prime }-f^{\prime \prime }\left (x \right ) y+f \left (x \right ) y^{3}-y^{4}=0} \]
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(diff(diff(y(x),x),x)*y(x)-diff(y(x),x)^2+diff(f(x),x)*diff(y(x),x)-diff(diff(f(x),x),x)*y(x)+f(x)*y(x)^3-y(x)^4=0,y(x), singsol=all)
\[ \text {No solution found} \]
✓ Solution by Mathematica
Time used: 60.651 (sec). Leaf size: 240
DSolve[f[x]*y[x]^3 - y[x]^4 + Derivative[1][f][x]*y'[x] - y'[x]^2 - y[x]*Derivative[2][f][x] + y[x]*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to -\frac {\exp \left (c_2-\int _1^x\frac {y(K[3])^4-f(K[3]) y(K[3])^3+\left (c_1+\int _1^{K[3]}\frac {-y(K[1])^4+f(K[1]) y(K[1])^3-f''(K[1]) y(K[1])+f'(K[1]) y'(K[1])}{y(K[1])^2}dK[1]\right ){}^2 y(K[3])^2+f''(K[3]) y(K[3])-f'(K[3]) y'(K[3])}{y(K[3])^2 \left (c_1+\int _1^{K[3]}\frac {-y(K[1])^4+f(K[1]) y(K[1])^3-f''(K[1]) y(K[1])+f'(K[1]) y'(K[1])}{y(K[1])^2}dK[1]\right )}dK[3]\right )}{\int _1^x\frac {-y(K[1])^4+f(K[1]) y(K[1])^3-f''(K[1]) y(K[1])+f'(K[1]) y'(K[1])}{y(K[1])^2}dK[1]+c_1} \]