61.29.19 problem 128

Internal problem ID [12549]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 2, Second-Order Differential Equations. section 2.1.2-4
Problem number : 128
Date solved : Monday, March 31, 2025 at 05:38:40 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

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

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

\[ y = \frac {c_1 \,{\mathrm e}^{a x} \left (a x -1\right )+c_2 \,{\mathrm e}^{-a x} \left (a x +1\right )}{x^{2}} \]

✓ Mathematica. Time used: 0.032 (sec). Leaf size: 29

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

\[ y(x)\to c_1 j_{-2}(i a x)-c_2 y_{-2}(i a x) \]

✓ Sympy. Time used: 0.237 (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)) + 2*x*Derivative(y(x), x) - (a**2*x**2 + 2)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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