23.3.486 problem 492

Internal problem ID [6200]
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 : 492
Date solved : Tuesday, September 30, 2025 at 02:36:13 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (c x +b \right ) y+a \,x^{2} y^{\prime }+x^{3} y^{\prime \prime }&=0 \end{align*}

✓ Maple. Time used: 0.008 (sec). Leaf size: 65

ode:=(c*x+b)*y(x)+a*x^2*diff(y(x),x)+x^3*diff(diff(y(x),x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.061 (sec). Leaf size: 133

ode=(b + c*x)*y[x] + a*x^2*D[y[x],x] + x^3*D[y[x],{x,2}] == 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
b = symbols("b") 
c = symbols("c") 
y = Function("y") 
ode = Eq(a*x**2*Derivative(y(x), x) + x**3*Derivative(y(x), (x, 2)) + (b + c*x)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

TypeError : invalid input: 1 - a