61.28.7 problem 67

Internal problem ID [12567]
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 : 67
Date solved : Tuesday, January 28, 2025 at 03:21:48 AM
CAS classification : [[_Emden, _Fowler]]

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

✓ Solution by Maple

Time used: 0.865 (sec). Leaf size: 71

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

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

✓ Solution by Mathematica

Time used: 0.139 (sec). Leaf size: 165

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

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