61.28.2 problem 62

Internal problem ID [12562]
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 : 62
Date solved : Tuesday, January 28, 2025 at 03:21:42 AM
CAS classification : [[_Emden, _Fowler]]

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

✓ Solution by Maple

Time used: 0.004 (sec). Leaf size: 83

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

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

✓ Solution by Mathematica

Time used: 0.071 (sec). Leaf size: 77

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

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