61.33.13 problem 251

Internal problem ID [12751]
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 : 251
Date solved : Tuesday, January 28, 2025 at 08:24:01 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.522 (sec). Leaf size: 65

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

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

✓ Solution by Mathematica

Time used: 0.108 (sec). Leaf size: 69

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

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