61.26.9 problem 9

Internal problem ID [12509]
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 : 9
Date solved : Tuesday, January 28, 2025 at 08:01:57 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.707 (sec). Leaf size: 113

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

\[ y = \frac {c_{2} x^{-\frac {3 n}{2}+\frac {1}{2}} \left (n -1\right )^{2} \operatorname {WhittakerM}\left (\frac {n -1}{2 n}, \frac {2 n -1}{2 n}, \frac {2 a \,x^{n}}{n}\right )}{2}+c_{2} \left (\frac {\left (n -1\right ) x^{-\frac {3 n}{2}+\frac {1}{2}}}{2}+a \,x^{-\frac {n}{2}+\frac {1}{2}}\right ) n \operatorname {WhittakerM}\left (-\frac {n +1}{2 n}, \frac {2 n -1}{2 n}, \frac {2 a \,x^{n}}{n}\right )+c_{1} x \,{\mathrm e}^{\frac {a \,x^{n}}{n}} \]

✗ Solution by Mathematica

Time used: 0.000 (sec). Leaf size: 0

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

Not solved