73.8.11 problem 13.2 (e)

Internal problem ID [15148]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 13. Higher order equations: Extending first order concepts. Additional exercises page 259
Problem number : 13.2 (e)
Date solved : Monday, March 31, 2025 at 01:29:18 PM
CAS classification : [[_2nd_order, _missing_y]]

\begin{align*} x y^{\prime \prime }-{y^{\prime }}^{2}&=6 x^{5} \end{align*}

✓ Maple. Time used: 0.019 (sec). Leaf size: 64

ode:=x*diff(diff(y(x),x),x)-diff(y(x),x)^2 = 6*x^5; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \sqrt {6}\, \int \frac {\left (\operatorname {BesselY}\left (1, \frac {2 x^{{5}/{2}} \sqrt {6}}{5}\right ) c_1 +\operatorname {BesselJ}\left (1, \frac {2 x^{{5}/{2}} \sqrt {6}}{5}\right )\right ) x^{{5}/{2}}}{c_1 \operatorname {BesselY}\left (0, \frac {2 x^{{5}/{2}} \sqrt {6}}{5}\right )+\operatorname {BesselJ}\left (0, \frac {2 x^{{5}/{2}} \sqrt {6}}{5}\right )}d x +c_2 \]

✓ Mathematica. Time used: 60.325 (sec). Leaf size: 109

ode=x*D[y[x],{x,2}]-D[y[x],x]^2==6*x^5; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \int _1^x\frac {\sqrt {6} \left (2 \operatorname {BesselY}\left (1,\frac {2}{5} \sqrt {6} K[1]^{5/2}\right )+\operatorname {BesselJ}\left (1,\frac {2}{5} \sqrt {6} K[1]^{5/2}\right ) c_1\right ) K[1]^{5/2}}{2 \operatorname {BesselY}\left (0,\frac {2}{5} \sqrt {6} K[1]^{5/2}\right )+\operatorname {BesselJ}\left (0,\frac {2}{5} \sqrt {6} K[1]^{5/2}\right ) c_1}dK[1]+c_2 \]

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-6*x**5 + x*Derivative(y(x), (x, 2)) - Derivative(y(x), x)**2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

ValueError : Rational Solution doesnt exist