61.26.7 problem 7

Internal problem ID [12507]
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 Equations Containing Power Functions. page 213
Problem number : 7
Date solved : Tuesday, January 28, 2025 at 03:19:02 AM
CAS classification : [[_Emden, _Fowler]]

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

✓ Solution by Maple

Time used: 1.180 (sec). Leaf size: 63

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

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

✓ Solution by Mathematica

Time used: 0.081 (sec). Leaf size: 119

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

\[ y(x)\to (n+2)^{-\frac {1}{n+2}} \sqrt {x} a^{\frac {1}{2 n+4}} \left (c_1 \operatorname {Gamma}\left (\frac {n+1}{n+2}\right ) \operatorname {BesselI}\left (-\frac {1}{n+2},\frac {2 \sqrt {a} x^{\frac {n}{2}+1}}{n+2}\right )+c_2 (-1)^{\frac {1}{n+2}} \operatorname {Gamma}\left (1+\frac {1}{n+2}\right ) \operatorname {BesselI}\left (\frac {1}{n+2},\frac {2 \sqrt {a} x^{\frac {n}{2}+1}}{n+2}\right )\right ) \]