61.28.3 problem 63

Internal problem ID [12563]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 2, Second-Order Differential Equations. section 2.1.2-3
Problem number : 63
Date solved : Tuesday, January 28, 2025 at 03:21:43 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x y^{\prime \prime }+a y^{\prime }+b x y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.004 (sec). Leaf size: 39

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

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

✓ Solution by Mathematica

Time used: 0.034 (sec). Leaf size: 54

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

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