23.3.196 problem 198

Internal problem ID [5910]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 3. THE DIFFERENTIAL EQUATION IS LINEAR AND OF SECOND ORDER, page 311
Problem number : 198
Date solved : Tuesday, September 30, 2025 at 02:06:05 PM
CAS classification : [[_Emden, _Fowler]]

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 83

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

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

✓ Mathematica. Time used: 0.037 (sec). Leaf size: 77

ode=b*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 b^{\frac {1}{2}-\frac {a}{2}} x^{\frac {1}{2}-\frac {a}{2}} \left (c_2 \operatorname {Gamma}(2-a) \operatorname {BesselJ}\left (1-a,2 \sqrt {b} \sqrt {x}\right )+c_1 \operatorname {Gamma}(a) \operatorname {BesselJ}\left (a-1,2 \sqrt {b} \sqrt {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*y(x) + x*Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

TypeError : invalid input: 1 - a