54.3.205 problem 1220

Internal problem ID [12500]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1220
Date solved : Friday, October 03, 2025 at 03:19:38 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.004 (sec). Leaf size: 38

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

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

✓ Mathematica. Time used: 0.224 (sec). Leaf size: 62

ode=y[x]*((1 - v)*v + x^2*(a + f[x]^2 + Derivative[1][f][x])) + 2*x^2*f[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 \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 ) \exp \left (\int _1^x\left (\frac {1}{2 K[1]}-f(K[1])\right )dK[1]\right ) \end{align*}

✗ Sympy

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

ValueError : Expected Expr or iterable but got None