Internal problem ID [11085]
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.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {x \left (x^{n}+1\right ) y^{\prime \prime }+\left (\left (a -b \right ) x^{n}+a -n \right ) y^{\prime }+b \left (1-a \right ) x^{-1+n} y=0} \]
✓ Solution by Maple
Time used: 0.032 (sec). Leaf size: 76
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)
\[ y \left (x \right ) = c_{1} \left (x^{n}+1\right )^{\frac {b}{n}}+c_{2} x^{-a +n +1} \left (x^{n}+1\right )^{\frac {b}{n}} \operatorname {hypergeom}\left (\left [\frac {b +n}{n}, \frac {-a +n +1}{n}\right ], \left [\frac {2 n -a +1}{n}\right ], -x^{n}\right ) \]
✓ Solution by Mathematica
Time used: 0.164 (sec). Leaf size: 69
DSolve[x*(x^n+1)*y''[x]+((a-b)*x^n+a-n)*y'[x]+b*(1-a)*x^(n-1)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to c_2 \left (x^n\right )^{\frac {-a+n+1}{n}} \operatorname {Hypergeometric2F1}\left (1,\frac {-a-b+n+1}{n},\frac {-a+2 n+1}{n},-x^n\right )+c_1 \left (x^n+1\right )^{b/n} \]