Internal problem ID [10089]
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: 8.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {y^{\prime \prime }-a \left (x^{2 n} a +x^{n -1} n \right ) y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 137
dsolve(diff(y(x),x$2)-a*(a*x^(2*n)+n*x^(n-1))*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = c_{1} {\mathrm e}^{\frac {x^{n +1} a}{n +1}}+c_{2} \left (\left (n +1\right ) \left (\left (\frac {n}{2}+1\right ) x^{-\frac {3 n}{2}-1}+x^{-\frac {n}{2}} a \right ) \WhittakerM \left (-\frac {n}{2 n +2}, \frac {2 n +3}{2 n +2}, \frac {2 x^{n +1} a}{n +1}\right )+\frac {x^{-\frac {3 n}{2}-1} \WhittakerM \left (\frac {n +2}{2 n +2}, \frac {2 n +3}{2 n +2}, \frac {2 x^{n +1} a}{n +1}\right ) \left (n +2\right )^{2}}{2}\right ) \]
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[y''[x]-a*(a*x^(2*n)+n*x^(n-1))*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
Not solved