61.31.1 problem 182

Internal problem ID [12603]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 2, Second-Order Differential Equations. section 2.1.2-6
Problem number : 182
Date solved : Wednesday, March 05, 2025 at 07:27:25 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.052 (sec). Leaf size: 49

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

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

✓ Mathematica. Time used: 0.09 (sec). Leaf size: 101

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

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

✓ Sympy. Time used: 0.168 (sec). Leaf size: 105

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

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