54.3.150 problem 1164

Internal problem ID [12445]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1164
Date solved : Wednesday, October 01, 2025 at 01:45:06 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 23

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

\[ y = c_1 \operatorname {BesselJ}\left (v , \sqrt {l}\, x \right )+c_2 \operatorname {BesselY}\left (v , \sqrt {l}\, x \right ) \]

✓ Mathematica. Time used: 0.017 (sec). Leaf size: 30

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

\begin{align*} y(x)&\to c_1 \operatorname {BesselJ}\left (v,\sqrt {l} x\right )+c_2 \operatorname {BesselY}\left (v,\sqrt {l} x\right ) \end{align*}

✓ Sympy. Time used: 0.148 (sec). Leaf size: 32

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

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