Internal problem ID [11098]
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: 264.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]
\[ \boxed {\left (x^{n} a +b \right )^{m +1} y^{\prime \prime }+\left (x^{n} a +b \right ) y^{\prime }-a n m \,x^{-1+n} y=0} \]
✗ Solution by Maple
dsolve((a*x^n+b)^(m+1)*diff(y(x),x$2)+(a*x^n+b)*diff(y(x),x)-a*n*m*x^(n-1)*y(x)=0,y(x), singsol=all)
\[ \text {No solution found} \]
✓ Solution by Mathematica
Time used: 0.249 (sec). Leaf size: 116
DSolve[(a*x^n+b)^(m+1)*y''[x]+(a*x^n+b)*y'[x]-a*n*m*x^(n-1)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \exp \left (-x \left (a x^n+b\right )^{-m} \left (\frac {a x^n}{b}+1\right )^m \operatorname {Hypergeometric2F1}\left (m,\frac {1}{n},1+\frac {1}{n},-\frac {a x^n}{b}\right )\right ) \left (\int _1^x\exp \left (\operatorname {Hypergeometric2F1}\left (m,\frac {1}{n},1+\frac {1}{n},-\frac {a K[1]^n}{b}\right ) K[1] \left (a K[1]^n+b\right )^{-m} \left (\frac {a K[1]^n}{b}+1\right )^m\right ) c_1dK[1]+c_2\right ) \]