61.29.16 problem 125

Internal problem ID [12546]
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 : 125
Date solved : Wednesday, March 05, 2025 at 07:09:34 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 19

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

\[ y = c_{1} \operatorname {BesselI}\left (n +\frac {1}{2}, x\right )+c_{2} \operatorname {BesselK}\left (n +\frac {1}{2}, x\right ) \]

✓ Mathematica. Time used: 0.032 (sec). Leaf size: 34

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

\[ y(x)\to c_1 \operatorname {BesselJ}\left (n+\frac {1}{2},-i x\right )+c_2 \operatorname {BesselY}\left (n+\frac {1}{2},-i x\right ) \]

✗ Sympy

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

False