Internal problem ID [10153]
Internal file name [OUTPUT/9100_Monday_June_06_2022_06_35_38_AM_96134006/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1832 (book 6.241).
ODE order: 2.
ODE degree: 2.
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.
\[ \boxed {F_{1,1}\left (x \right ) {y^{\prime }}^{2}+\left (\left (F_{2,1}\left (x \right )+F_{1,2}\left (x \right )\right ) y^{\prime \prime }+y \left (F_{1,0}\left (x \right )+F_{0,1}\left (x \right )\right )\right ) y^{\prime }+F_{2,2}\left (x \right ) {y^{\prime \prime }}^{2}+y \left (F_{2,0}\left (x \right )+F_{0,2}\left (x \right )\right ) y^{\prime \prime }+F_{0,0}\left (x \right ) y^{2}=0} \] Does not support ODE with \({y^{\prime \prime }}^{n}\) where \(n\neq 1\) unless \(F_{0,0}\left (x \right ) y\) is missing which is not the case here.
Maple trace
`Methods for second order ODEs: Successful isolation of d^2y/dx^2: 2 solutions were found. Trying to solve each resulting ODE. *** Sublevel 2 *** Methods for second order ODEs: --- Trying classification methods --- 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 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) `, `-> 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) = -(1/2)*(2*F[2, 2](_a)*_b(_a)^2+_b(_a)*F[2, 1](_a)+_b(_a)*F[1, 2](_a)-(-4*F[0 Methods for first order ODEs: --- Trying classification methods --- trying homogeneous types: trying exact Looking for potential symmetries trying an equivalence to an Abel ODE 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)* solving 2nd order ODE of high degree, Lie methods `, `-> Computing symmetries using: way = 3`[0, y]
✗ Solution by Maple
dsolve(F[1,1](x)*diff(y(x),x)^2+((F[2,1](x)+F[1,2](x))*diff(diff(y(x),x),x)+y(x)*(F[1,0](x)+F[0,1](x)))*diff(y(x),x)+F[2,2](x)*diff(diff(y(x),x),x)^2+y(x)*(F[2,0](x)+F[0,2](x))*diff(diff(y(x),x),x)+F[0,0](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[x*F[0, 0]*y[x]^2 + x*F[1, 1]*y''[x] + (x*F[0, 2] + x*F[2, 0])*y[x]*y''[x] + x*F[2, 2]*y''[x]^2 + y'[x]*((x*F[0, 1] + x*F[1, 0])*y[x] + (x*F[1, 2] + x*F[2, 1])*y''[x]) == 0,y[x],x,IncludeSingularSolutions -> True]
Not solved