61.26.8 problem 8

Internal problem ID [12508]
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 : 8
Date solved : Tuesday, January 28, 2025 at 08:01:57 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.789 (sec). Leaf size: 136

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

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

✓ Solution by Mathematica

Time used: 0.356 (sec). Leaf size: 81

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

\[ y(x)\to e^{\frac {a x^{n+1}}{n+1}} \left (c_2-\frac {c_1 2^{-\frac {1}{n+1}} x \left (\frac {a x^{n+1}}{n+1}\right )^{-\frac {1}{n+1}} \Gamma \left (\frac {1}{n+1},\frac {2 a x^{n+1}}{n+1}\right )}{n+1}\right ) \]