54.3.148 problem 1162

Internal problem ID [12443]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1162
Date solved : Wednesday, October 01, 2025 at 01:45:05 AM
CAS classification : [_Bessel]

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 15

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

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

✓ Mathematica. Time used: 0.034 (sec). Leaf size: 18

ode=(-v^2 + 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}(v,x)+c_2 \operatorname {BesselY}(v,x) \end{align*}

✓ Sympy. Time used: 0.146 (sec). Leaf size: 22

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

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