Internal problem ID [10928]
Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev.
Second edition
Section: Chapter 2, Second-Order Differential Equations. section 2.1.2-3 Equation of form
\((a x + b)y''+f(x)y'+g(x)y=0\)
Problem number: 94.
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)]`]]
\[ \boxed {x y^{\prime \prime }+\left (x^{n} a +b \right ) y^{\prime }+x^{-1+n} y a n=0} \]
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 62
dsolve(x*diff(y(x),x$2)+(a*x^n+b)*diff(y(x),x)+a*n*x^(n-1)*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = c_{1} {\mathrm e}^{-\frac {a \,x^{n}}{n}} \operatorname {hypergeom}\left (\left [\frac {b -1}{n}\right ], \left [\frac {n +b -1}{n}\right ], \frac {a \,x^{n}}{n}\right )+c_{2} x^{-b +1} {\mathrm e}^{-\frac {a \,x^{n}}{n}} \]
✓ Solution by Mathematica
Time used: 0.244 (sec). Leaf size: 121
DSolve[x*y''[x]+(a*x^n+b)*y'[x]+a*n*x^(n-1)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to (-1)^{-\frac {b}{n}} n^{\frac {b-n-1}{n}} a^{\frac {1-b}{n}} e^{-\frac {a x^n}{n}} \left (x^n\right )^{\frac {1-b}{n}} \left (-(b-1) c_1 (-1)^{\frac {1}{n}} \Gamma \left (\frac {b-1}{n},-\frac {a x^n}{n}\right )+c_2 n (-1)^{b/n}+(b-1) c_1 (-1)^{\frac {1}{n}} \operatorname {Gamma}\left (\frac {b-1}{n}\right )\right ) \]