61.29.2 problem 111

Internal problem ID [12532]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 2, Second-Order Differential Equations. section 2.1.2-4
Problem number : 111
Date solved : Monday, March 31, 2025 at 05:38:10 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 45

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

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

✓ Mathematica. Time used: 0.098 (sec). Leaf size: 95

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

\[ y(x)\to \sqrt {a} \sqrt {x} \left (c_1 \operatorname {Gamma}\left (1-\sqrt {1-4 b}\right ) \operatorname {BesselJ}\left (-\sqrt {1-4 b},2 \sqrt {a} \sqrt {x}\right )+c_2 \operatorname {Gamma}\left (\sqrt {1-4 b}+1\right ) \operatorname {BesselJ}\left (\sqrt {1-4 b},2 \sqrt {a} \sqrt {x}\right )\right ) \]

✓ Sympy. Time used: 0.130 (sec). Leaf size: 54

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

\[ y{\left (x \right )} = \sqrt {x} \left (C_{1} J_{2 \sqrt {\frac {1}{4} - b}}\left (2 \sqrt {a} \sqrt {x}\right ) + C_{2} Y_{2 \sqrt {\frac {1}{4} - b}}\left (2 \sqrt {a} \sqrt {x}\right )\right ) \]