61.29.15 problem 124

Internal problem ID [12624]
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 : 124
Date solved : Tuesday, January 28, 2025 at 03:23:26 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 19

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

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

✓ Solution by Mathematica

Time used: 0.229 (sec). Leaf size: 26

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

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