23.3.249 problem 251

Internal problem ID [5963]
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 : 251
Date solved : Tuesday, September 30, 2025 at 02:07:00 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 45

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

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

✓ Mathematica. Time used: 0.051 (sec). Leaf size: 95

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

\begin{align*} y(x)&\to \sqrt {b} \sqrt {x} \left (c_1 \operatorname {Gamma}\left (1-\sqrt {1-4 a}\right ) \operatorname {BesselJ}\left (-\sqrt {1-4 a},2 \sqrt {b} \sqrt {x}\right )+c_2 \operatorname {Gamma}\left (\sqrt {1-4 a}+1\right ) \operatorname {BesselJ}\left (\sqrt {1-4 a},2 \sqrt {b} \sqrt {x}\right )\right ) \end{align*}

✓ Sympy. Time used: 0.063 (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 + b*x)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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