61.28.2 problem 62

Internal problem ID [12483]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 2, Second-Order Differential Equations. section 2.1.2-3
Problem number : 62
Date solved : Monday, March 31, 2025 at 05:36:10 AM
CAS classification : [[_Emden, _Fowler]]

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 83

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

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

✓ Mathematica. Time used: 0.071 (sec). Leaf size: 77

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

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

✗ 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