23.3.252 problem 254

Internal problem ID [5966]
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 : 254
Date solved : Tuesday, September 30, 2025 at 02:07:02 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.005 (sec). Leaf size: 32

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

\[ y = \frac {c_1 \,{\mathrm e}^{a x} \left (a x -1\right )+c_2 \,{\mathrm e}^{-a x} \left (a x +1\right )}{x} \]

✓ Mathematica. Time used: 0.046 (sec). Leaf size: 67

ode=-((2 + 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 }} a x^{3/2} ((i a c_2 x+c_1) \sinh (a x)-(a c_1 x+i c_2) \cosh (a x))}{(-i a x)^{5/2}} \end{align*}

✓ Sympy. Time used: 0.051 (sec). Leaf size: 37

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 - 2)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \sqrt {x} \left (C_{1} J_{\frac {3}{2}}\left (x \sqrt {- a^{2}}\right ) + C_{2} Y_{\frac {3}{2}}\left (x \sqrt {- a^{2}}\right )\right ) \]