Internal problem ID [10091]
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: 10.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {y^{\prime \prime }+\left (x^{2 n} a +x^{n -1} b \right ) y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.258 (sec). Leaf size: 95
dsolve(diff(y(x),x$2)+(a*x^(2*n)+b*x^(n-1))*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = c_{1} \WhittakerM \left (-\frac {i b}{\sqrt {a}\, \left (2 n +2\right )}, \frac {1}{2 n +2}, \frac {2 i \sqrt {a}\, x^{n +1}}{n +1}\right ) x^{-\frac {n}{2}}+c_{2} \WhittakerW \left (-\frac {i b}{\sqrt {a}\, \left (2 n +2\right )}, \frac {1}{2 n +2}, \frac {2 i \sqrt {a}\, x^{n +1}}{n +1}\right ) x^{-\frac {n}{2}} \]
✓ Solution by Mathematica
Time used: 0.075 (sec). Leaf size: 199
DSolve[y''[x]+(a*x^(2*n)+b*x^(n-1))*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to 2^{\frac {n}{2 n+2}} x^{-n/2} \left (x^{n+1}\right )^{\frac {n}{2 n+2}} e^{-\frac {\sqrt {a} x^{n+1}}{\sqrt {-(n+1)^2}}} \left (c_1 \text {HypergeometricU}\left (\frac {-\frac {b \sqrt {-(n+1)^2}}{\sqrt {a}}+n^2+n}{2 (n+1)^2},\frac {n}{n+1},\frac {2 \sqrt {a} x^{n+1}}{\sqrt {-(n+1)^2}}\right )+c_2 \text {LaguerreL}\left (-\frac {-\frac {b \sqrt {-(n+1)^2}}{\sqrt {a}}+n^2+n}{2 (n+1)^2},-\frac {1}{n+1},\frac {2 \sqrt {a} x^{n+1}}{\sqrt {-(n+1)^2}}\right )\right ) \\ \end{align*}