Internal problem ID [10156]
Internal file name [OUTPUT/9103_Monday_June_06_2022_06_37_05_AM_82327004/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1835 (book 6.244).
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.
\[ \boxed {\left (y^{2}-x^{2} {y^{\prime }}^{2}+x^{2} y^{\prime \prime } y\right )^{2}-4 x y \left (y^{\prime } x -y\right )^{3}=0} \] Does not support ODE with \({y^{\prime \prime }}^{n}\) where \(n\neq 1\) unless \(4 y^{3} x +y^{3}\) is missing which is not the case here.
Maple trace
`Methods for second order ODEs: *** Sublevel 2 *** Methods for second order ODEs: Successful isolation of d^2y/dx^2: 2 solutions were found. Trying to solve each resulting ODE. *** Sublevel 3 *** 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) 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) = (2*(_a*(_b(_a)*_a-1))^(1/2)*_b(_a)*_a-2*(_a*(_b(_a)*_a-1))^(1/2)-1)/_a^2, 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, solving 2nd order ODE of high degree, Lie methods `, `-> Computing symmetries using: way = 3`[0, y]
✗ Solution by Maple
dsolve((y(x)^2-x^2*diff(y(x),x)^2+x^2*y(x)*diff(diff(y(x),x),x))^2-4*x*y(x)*(x*diff(y(x),x)-y(x))^3=0,y(x), singsol=all)
\[ \text {No solution found} \]
✓ Solution by Mathematica
Time used: 96.129 (sec). Leaf size: 27
DSolve[-4*x*y[x]*(-y[x] + x*y'[x])^3 + (y[x]^2 - x^2*y'[x]^2 + x^2*y[x]*y''[x])^2 == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to c_1 x e^{\frac {1}{-x+c_2}} \\ y(x)\to c_1 x \\ \end{align*}