Internal problem ID [10009]
Internal file name [OUTPUT/8956_Monday_June_06_2022_06_03_11_AM_47071803/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1688 (book 6.97).
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, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]
Unable to solve or complete the solution.
\[ \boxed {x^{4} y^{\prime \prime }-x \left (x^{2}+2 y\right ) y^{\prime }+4 y^{2}=0} \]
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) -> Calling odsolve with the ODE`, (diff(_b(_a), _a))/_a-(-c__1*_a^4+_b(_a)^2)/_a^4 = 0, _b(_a)` *** Sublevel 2 *** 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 homogeneous G <- homogeneous successful <- differential order: 2; exact nonlinear successful`
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 22
dsolve(x^4*diff(diff(y(x),x),x)-x*(x^2+2*y(x))*diff(y(x),x)+4*y(x)^2=0,y(x), singsol=all)
\[ y \left (x \right ) = \left (-\tanh \left (c_{1} \left (\ln \left (x \right )-c_{2} \right )\right ) c_{1} +1\right ) x^{2} \]
✓ Solution by Mathematica
Time used: 79.662 (sec). Leaf size: 83
DSolve[4*y[x]^2 - x*(x^2 + 2*y[x])*y'[x] + x^4*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {x^2 \left (\left (1-i \sqrt {-1-c_1}\right ) x^{2 i \sqrt {-1-c_1}}+\left (1+i \sqrt {-1-c_1}\right ) c_2\right )}{x^{2 i \sqrt {-1-c_1}}+c_2} \]