54.3.139 problem 1153

Internal problem ID [12434]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1153
Date solved : Wednesday, October 01, 2025 at 01:44:58 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 31

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

\[ y = \left (\operatorname {BesselJ}\left (v -\frac {1}{2}, \sqrt {a}\, x \right ) c_1 +\operatorname {BesselY}\left (v -\frac {1}{2}, \sqrt {a}\, x \right ) c_2 \right ) \sqrt {x} \]

✓ Mathematica. Time used: 0.021 (sec). Leaf size: 44

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

\begin{align*} y(x)&\to \sqrt {x} \left (c_1 \operatorname {BesselJ}\left (v-\frac {1}{2},\sqrt {a} x\right )+c_2 \operatorname {BesselY}\left (v-\frac {1}{2},\sqrt {a} x\right )\right ) \end{align*}

✓ Sympy. Time used: 0.069 (sec). Leaf size: 48

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

\[ y{\left (x \right )} = \sqrt {x} \left (C_{1} J_{\sqrt {v \left (v - 1\right ) + \frac {1}{4}}}\left (\sqrt {a} x\right ) + C_{2} Y_{\sqrt {v \left (v - 1\right ) + \frac {1}{4}}}\left (\sqrt {a} x\right )\right ) \]