61.26.10 problem 10

Internal problem ID [12510]
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
Date solved : Tuesday, January 28, 2025 at 08:01:58 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+\left (a \,x^{2 n}+b \,x^{n -1}\right ) y&=0 \end{align*}

✓ Solution by Maple

Time used: 1.576 (sec). Leaf size: 85

dsolve(diff(y(x),x$2)+(a*x^(2*n)+b*x^(n-1))*y(x)=0,y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.239 (sec). Leaf size: 225

DSolve[D[y[x],{x,2}]+(a*x^(2*n)+b*x^(n-1))*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to 2^{\frac {n}{2 n+2}} x^{-n/2} \left (x^{n+1}\right )^{\frac {n}{2 n+2}} e^{-\frac {\sqrt {a} x^{n+1}}{\sqrt {-(n+1)^2}}} \left (c_1 \operatorname {HypergeometricU}\left (-\frac {(n+1) \left (n b+b+\sqrt {a} n \sqrt {-(n+1)^2}\right )}{2 \sqrt {a} \left (-(n+1)^2\right )^{3/2}},\frac {n}{n+1},\frac {2 \sqrt {a} x^{n+1}}{\sqrt {-(n+1)^2}}\right )+c_2 L_{\frac {(n+1) \left (n b+b+\sqrt {a} n \sqrt {-(n+1)^2}\right )}{2 \sqrt {a} \left (-(n+1)^2\right )^{3/2}}}^{-\frac {1}{n+1}}\left (\frac {2 \sqrt {a} x^{n+1}}{\sqrt {-(n+1)^2}}\right )\right ) \]