23.3.253 problem 255

Internal problem ID [5967]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 3. THE DIFFERENTIAL EQUATION IS LINEAR AND OF SECOND ORDER, page 311
Problem number : 255
Date solved : Tuesday, September 30, 2025 at 02:07:03 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} -\left (-a^{2} x^{2}+6\right ) y+x^{2} y^{\prime \prime }&=0 \end{align*}

✓ Maple. Time used: 0.032 (sec). Leaf size: 53

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

\[ y = \frac {\left (c_1 \,a^{2} x^{2}+3 c_2 a x -3 c_1 \right ) \cos \left (a x \right )+\sin \left (a x \right ) \left (c_2 \,a^{2} x^{2}-3 c_1 a x -3 c_2 \right )}{x^{2}} \]

✓ Mathematica. Time used: 0.056 (sec). Leaf size: 79

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

\begin{align*} y(x)&\to -\frac {\sqrt {\frac {2}{\pi }} \sqrt {x} \left (\left (-a^2 c_2 x^2+3 a c_1 x+3 c_2\right ) \cos (a x)+\left (c_1 \left (a^2 x^2-3\right )+3 a c_2 x\right ) \sin (a x)\right )}{(a x)^{5/2}} \end{align*}

✓ Sympy. Time used: 0.046 (sec). Leaf size: 24

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

\[ y{\left (x \right )} = \sqrt {x} \left (C_{1} J_{\frac {5}{2}}\left (a x\right ) + C_{2} Y_{\frac {5}{2}}\left (a x\right )\right ) \]