54.3.91 problem 1105

Internal problem ID [12386]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1105
Date solved : Wednesday, October 01, 2025 at 01:44:12 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x y^{\prime \prime }+a y^{\prime }+b x y&=0 \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 39

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

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

✓ Mathematica. Time used: 0.021 (sec). Leaf size: 54

ode=b*x*y[x] + a*D[y[x],x] + x*D[y[x],{x,2}] == 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✗ Sympy

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

TypeError : invalid input: 1 - a