61.33.15 problem 253

Internal problem ID [12753]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 2, Second-Order Differential Equations. section 2.1.2-8. Other equations.
Problem number : 253
Date solved : Tuesday, January 28, 2025 at 08:24:03 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.502 (sec). Leaf size: 23

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

\[ y = c_{1} \operatorname {LegendreP}\left (\nu , a \,x^{n}\right )+c_{2} \operatorname {LegendreQ}\left (\nu , a \,x^{n}\right ) \]

✓ Solution by Mathematica

Time used: 0.121 (sec). Leaf size: 79

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

\[ y(x)\to i a c_2 \sqrt {x^{2 n}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2}-\frac {\nu }{2},\frac {\nu }{2}+1,\frac {3}{2},a^2 x^{2 n}\right )+c_1 \operatorname {Hypergeometric2F1}\left (-\frac {\nu }{2},\frac {\nu +1}{2},\frac {1}{2},a^2 x^{2 n}\right ) \]