Internal problem ID [10011]
Internal file name [OUTPUT/8958_Monday_June_06_2022_06_03_21_AM_39336745/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1690 (book 6.99).
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]]
Unable to solve or complete the solution.
\[ \boxed {x^{4} y^{\prime \prime }+\left (x y^{\prime }-y\right )^{3}=0} \]
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 a change of variables {x -> y(x), y(x) -> x} 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) `, `-> Computing symmetries using: way = 3 Try integration with the canonical coordinates of the symmetry [0, x] -> Calling odsolve with the ODE`, diff(_b(_a), _a) = -_b(_a)*(_b(_a)^2*_a^2+2)/_a, _b(_a), explicit, HINT = [[_a, -_b]]` *** Suble symmetry methods on request `, `1st order, trying reduction of order with given symmetries:`[_a, -_b]
✓ Solution by Maple
Time used: 0.047 (sec). Leaf size: 37
dsolve(x^4*diff(diff(y(x),x),x)+(x*diff(y(x),x)-y(x))^3=0,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= \left (-\arctan \left (\frac {1}{\sqrt {c_{1} x^{2}-1}}\right )+c_{2} \right ) x \\ y \left (x \right ) &= \left (\arctan \left (\frac {1}{\sqrt {c_{1} x^{2}-1}}\right )+c_{2} \right ) x \\ \end{align*}
✓ Solution by Mathematica
Time used: 60.327 (sec). Leaf size: 95
DSolve[(-y[x] + x*y'[x])^3 + x^4*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -i x \log \left (\frac {e^{c_2}-\sqrt {e^{2 c_2}-8 i c_1 x^2}}{4 c_1 x}\right ) \\ y(x)\to -i x \log \left (\frac {\sqrt {e^{2 c_2}-8 i c_1 x^2}+e^{c_2}}{4 c_1 x}\right ) \\ \end{align*}