61.29.2 problem 111

Internal problem ID [12611]
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 : Tuesday, January 28, 2025 at 03:23:07 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.005 (sec). Leaf size: 45

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

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

✓ Solution by Mathematica

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

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

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