Internal problem ID [10088]
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.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_Emden, _Fowler]]
Solve \begin {gather*} \boxed {y^{\prime \prime }-a \,x^{n} y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.202 (sec). Leaf size: 65
dsolve(diff(y(x),x$2)-a*x^n*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = c_{1} \sqrt {x}\, \BesselJ \left (\frac {1}{n +2}, \frac {2 \sqrt {-a}\, x^{\frac {n}{2}+1}}{n +2}\right )+c_{2} \sqrt {x}\, \BesselY \left (\frac {1}{n +2}, \frac {2 \sqrt {-a}\, x^{\frac {n}{2}+1}}{n +2}\right ) \]
✓ Solution by Mathematica
Time used: 0.013 (sec). Leaf size: 119
DSolve[y''[x]-a*x^n*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to (n+2)^{-\frac {1}{n+2}} \sqrt {x} a^{\frac {1}{2 n+4}} \left (c_1 \text {Gamma}\left (\frac {n+1}{n+2}\right ) I_{-\frac {1}{n+2}}\left (\frac {2 \sqrt {a} x^{\frac {n}{2}+1}}{n+2}\right )+c_2 (-1)^{\frac {1}{n+2}} \text {Gamma}\left (\frac {1}{n+2}+1\right ) I_{\frac {1}{n+2}}\left (\frac {2 \sqrt {a} x^{\frac {n}{2}+1}}{n+2}\right )\right ) \\ \end{align*}