Internal problem ID [12202]
Internal file name [OUTPUT/10855_Thursday_September_21_2023_05_48_08_AM_58249007/index.tex]
Book: Differential equations and the calculus of variations by L. ElSGOLTS. MIR PUBLISHERS,
MOSCOW, Third printing 1977.
Section: Chapter 2, DIFFERENTIAL EQUATIONS OF THE SECOND ORDER AND HIGHER.
Problems page 172
Problem number: Problem 55.
ODE order: 1.
ODE degree: 2.
The type(s) of ODE detected by this program : "unknown"
Maple gives the following as the ode type
[[_high_order, _missing_x], [_high_order, _missing_y], [_high_order, _with_linear_symmetries], [_high_order, _reducible, _mu_poly_yn]]
Unable to solve or complete the solution.
Unable to parse ODE.
Maple trace
`Methods for high order ODEs: --- Trying classification methods --- trying 4th order ODE linearizable_by_differentiation trying high order reducible trying differential order: 4; mu polynomial in y -> Calling odsolve with the ODE`, -(5/6)*ln(diff(diff(_b(_a), _a), _a))+ln(diff(diff(diff(_b(_a), _a), _a), _a))+c__1 = 0, _b(_a)` Methods for third order ODEs: --- Trying classification methods --- trying 3rd order ODE linearizable_by_differentiation differential order: 3; trying a linearization to 4th order trying differential order: 3; missing variables `, `-> Computing symmetries using: way = 3 -> Calling odsolve with the ODE`, diff(_g(_f), _f) = _g(_f)^(5/6)*exp(-c__1), _g(_f), HINT = [[1, 0], [_f, 6*_g]]` *** Suble symmetry methods on request `, `1st order, trying reduction of order with given symmetries:`[1, 0], [_f, 6*_g]
✓ Solution by Maple
Time used: 0.062 (sec). Leaf size: 25
dsolve(6*diff(y(x),x$2)*diff(y(x),x$4)-5*diff(y(x),x$3)^2=0,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= c_{1} x +c_{2} \\ y \left (x \right ) &= \frac {\left (c_{2} +x \right )^{8} c_{1}}{2612736}+c_{3} x +c_{4} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.266 (sec). Leaf size: 26
DSolve[6*y''[x]*y''''[x]-5*y'''[x]^2==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {1}{56} c_2 (x-6 c_1){}^8+c_4 x+c_3 \]