23.3.313 problem 315

Internal problem ID [6027]
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 : 315
Date solved : Tuesday, September 30, 2025 at 02:20:07 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 65

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

\[ y = x^{-\frac {a}{2}+\frac {1}{2}} \left (\operatorname {BesselY}\left (\frac {\sqrt {a^{2}-2 a -4 b +1}}{3}, \frac {2 \sqrt {c}\, x^{{3}/{2}}}{3}\right ) c_2 +\operatorname {BesselJ}\left (\frac {\sqrt {a^{2}-2 a -4 b +1}}{3}, \frac {2 \sqrt {c}\, x^{{3}/{2}}}{3}\right ) c_1 \right ) \]

✓ Mathematica. Time used: 0.083 (sec). Leaf size: 156

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

\begin{align*} y(x)&\to 3^{\frac {a-1}{3}} c^{\frac {1}{6}-\frac {a}{6}} x^{\frac {1}{2}-\frac {a}{2}} \left (c_1 \operatorname {Gamma}\left (1-\frac {1}{3} \sqrt {a^2-2 a-4 b+1}\right ) \operatorname {BesselJ}\left (-\frac {1}{3} \sqrt {a^2-2 a-4 b+1},\frac {2}{3} \sqrt {c} x^{3/2}\right )+c_2 \operatorname {Gamma}\left (\frac {1}{3} \sqrt {a^2-2 a-4 b+1}+1\right ) \operatorname {BesselJ}\left (\frac {1}{3} \sqrt {a^2-2 a-4 b+1},\frac {2}{3} \sqrt {c} x^{3/2}\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*Derivative(y(x), x) + x**2*Derivative(y(x), (x, 2)) + (b + c*x**3)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

TypeError : invalid input: 1 - a