87.14.11 problem 11

Internal problem ID [23530]
Book : Ordinary differential equations with modern applications. Ladas, G. E. and Finizio, N. Wadsworth Publishing. California. 1978. ISBN 0-534-00552-7. QA372.F56
Section : Chapter 2. Linear differential equations. Exercise at page 109
Problem number : 11
Date solved : Thursday, October 02, 2025 at 09:42:44 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+\frac {y^{\prime }}{x^{{1}/{3}}}+\left (\frac {1}{4 x^{{2}/{3}}}-\frac {1}{6 x^{{4}/{3}}}-\frac {6}{x^{2}}\right ) y&=0 \end{align*}

Using reduction of order method given that one solution is

\begin{align*} y&=x^{3} {\mathrm e}^{-\frac {3 x^{{2}/{3}}}{4}} \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 21

ode:=diff(diff(y(x),x),x)+1/x^(1/3)*diff(y(x),x)+(1/4/x^(2/3)-1/6/x^(4/3)-6/x^2)*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {{\mathrm e}^{-\frac {3 x^{{2}/{3}}}{4}} \left (c_2 \,x^{5}+c_1 \right )}{x^{2}} \]

✓ Mathematica. Time used: 0.041 (sec). Leaf size: 34

ode=D[y[x],{x,2}]+1/x^(1/3)*D[y[x],x]+(1/4*x^(-2/3)-1/6*x^(-4/3)-6*x^(-2))*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {e^{-\frac {3 x^{2/3}}{4}} \left (c_2 x^5+5 c_1\right )}{5 x^2} \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((-6/x**2 + 1/(4*x**(2/3)) - 1/(6*x**(4/3)))*y(x) + Derivative(y(x), (x, 2)) + Derivative(y(x), x)/x**(1/3),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE x**(1/3)*Derivative(y(x), (x, 2)) + Derivative(y(x), x) - y(x)/(