54.3.192 problem 1206

Internal problem ID [12487]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1206
Date solved : Friday, October 03, 2025 at 03:19:23 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }+\left (2 a x +b \right ) x y^{\prime }+\left (a b x +c \,x^{2}+d \right ) y&=0 \end{align*}

✓ Maple. Time used: 0.005 (sec). Leaf size: 76

ode:=x^2*diff(diff(y(x),x),x)+(2*a*x+b)*x*diff(y(x),x)+(a*b*x+c*x^2+d)*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = x^{-\frac {b}{2}+\frac {1}{2}} {\mathrm e}^{-a x} \left (\operatorname {BesselY}\left (\frac {\sqrt {b^{2}-2 b -4 d +1}}{2}, \sqrt {-a^{2}+c}\, x \right ) c_2 +\operatorname {BesselJ}\left (\frac {\sqrt {b^{2}-2 b -4 d +1}}{2}, \sqrt {-a^{2}+c}\, x \right ) c_1 \right ) \]

✓ Mathematica. Time used: 0.085 (sec). Leaf size: 110

ode=(d + a*b*x + c*x^2)*y[x] + x*(b + 2*a*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 (\frac {1}{2} \sqrt {b^2-2 b-4 d+1},-i \sqrt {a^2-c} x\right )+c_2 \operatorname {BesselY}\left (\frac {1}{2} \sqrt {b^2-2 b-4 d+1},-i \sqrt {a^2-c} x\right )\right ) \exp \left (\int _1^x-\frac {b+2 a K[1]-1}{2 K[1]}dK[1]\right ) \end{align*}

✗ Sympy

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

ValueError : Expected Expr or iterable but got None