61.29.18 problem 127

Internal problem ID [12548]
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 : 127
Date solved : Wednesday, March 05, 2025 at 07:09:37 PM
CAS classification : [[_Bessel, _modified]]

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 15

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

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

✓ Mathematica. Time used: 0.034 (sec). Leaf size: 34

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

\[ y(x)\to c_1 \operatorname {BesselJ}\left (\sqrt {\nu },-i x\right )+c_2 \operatorname {BesselY}\left (\sqrt {\nu },-i x\right ) \]

✓ Sympy. Time used: 0.227 (sec). Leaf size: 26

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

\[ y{\left (x \right )} = C_{1} J_{\sqrt {\nu ^{2}}}\left (i x\right ) + C_{2} Y_{\sqrt {\nu ^{2}}}\left (i x\right ) \]