54.3.206 problem 1221

Internal problem ID [12501]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1221
Date solved : Friday, October 03, 2025 at 03:19:40 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.005 (sec). Leaf size: 35

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

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

✓ Mathematica. Time used: 0.128 (sec). Leaf size: 31

ode=y[x]*(-v^2 - x*f[x] + x^2*(1 + f[x]^2 - Derivative[1][f][x])) + (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 (c_1 \operatorname {BesselJ}(v,x)+c_2 \operatorname {BesselY}(v,x)) \exp \left (\int _1^xf(K[1])dK[1]\right ) \end{align*}

✗ Sympy

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

ValueError : Expected Expr or iterable but got None