61.29.24 problem 133

Internal problem ID [12633]
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 : 133
Date solved : Tuesday, January 28, 2025 at 08:02:55 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.684 (sec). Leaf size: 78

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

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

✓ Solution by Mathematica

Time used: 0.153 (sec). Leaf size: 165

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

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