61.29.11 problem 120

Internal problem ID [12620]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 2, Second-Order Differential Equations. section 2.1.2-4
Problem number : 120
Date solved : Tuesday, January 28, 2025 at 08:02:53 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.631 (sec). Leaf size: 84

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

\[ y = x^{-\frac {n}{2}} \sqrt {x}\, \left (\operatorname {WhittakerM}\left (-\frac {i b}{2 n \sqrt {a}}, \frac {i \sqrt {4 c -1}}{2 n}, \frac {2 i \sqrt {a}\, x^{n}}{n}\right ) c_{1} +\operatorname {WhittakerW}\left (-\frac {i b}{2 n \sqrt {a}}, \frac {i \sqrt {4 c -1}}{2 n}, \frac {2 i \sqrt {a}\, x^{n}}{n}\right ) c_{2} \right ) \]

✓ Solution by Mathematica

Time used: 0.175 (sec). Leaf size: 236

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

\[ y(x)\to 2^{\frac {\sqrt {(1-4 c) n^2}+n^2}{2 n^2}} x^{\frac {1}{2}-\frac {n}{2}} e^{\frac {i \sqrt {a} x^n}{n}} \left (x^n\right )^{\frac {\sqrt {(1-4 c) n^2}+n^2}{2 n^2}} \left (c_1 \operatorname {HypergeometricU}\left (\frac {1}{2} \left (-\frac {i b}{\sqrt {a} n}+\frac {\sqrt {(1-4 c) n^2}}{n^2}+1\right ),\frac {\sqrt {(1-4 c) n^2}}{n^2}+1,-\frac {2 i \sqrt {a} x^n}{n}\right )+c_2 L_{\frac {1}{2} \left (\frac {i b}{\sqrt {a} n}-\frac {\sqrt {(1-4 c) n^2}}{n^2}-1\right )}^{\frac {\sqrt {(1-4 c) n^2}}{n^2}}\left (-\frac {2 i \sqrt {a} x^n}{n}\right )\right ) \]