61.29.3 problem 112

Internal problem ID [12612]
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 : 112
Date solved : Tuesday, January 28, 2025 at 03:23:08 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }+\left (a^{2} x^{2}-n \left (n +1\right )\right ) y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.004 (sec). Leaf size: 27

dsolve(x^2*diff(y(x),x$2)+(a^2*x^2-n*(n+1))*y(x)=0,y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.035 (sec). Leaf size: 36

DSolve[x^2*D[y[x],{x,2}]+(a^2*x^2-n*(n+1))*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \sqrt {x} \left (c_1 \operatorname {BesselJ}\left (n+\frac {1}{2},a x\right )+c_2 \operatorname {BesselY}\left (n+\frac {1}{2},a x\right )\right ) \]