Internal problem ID [10930]
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: 96.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {x y^{\prime \prime }+\left (x^{n} a +b \right ) y^{\prime }+a \left (-1+b +n \right ) x^{-1+n} y=0} \]
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 66
dsolve(x*diff(y(x),x$2)+(a*x^n+b)*diff(y(x),x)+a*(b+n-1)*x^(n-1)*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = c_{1} {\mathrm e}^{-\frac {a \,x^{n}}{n}}+c_{2} x^{-b +1} {\mathrm e}^{-\frac {a \,x^{n}}{n}} \operatorname {hypergeom}\left (\left [\frac {-b +1}{n}\right ], \left [\frac {-b +n +1}{n}\right ], \frac {a \,x^{n}}{n}\right ) \]
✓ Solution by Mathematica
Time used: 0.178 (sec). Leaf size: 93
DSolve[x*y''[x]+(a*x^n+b)*y'[x]+a*(b+n-1)*x^(n-1)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {(-1)^{-1/n} e^{-\frac {a x^n}{n}} \left ((b-1) c_2 (-1)^{b/n} \Gamma \left (\frac {1-b}{n},-\frac {a x^n}{n}\right )-(b-1) c_2 (-1)^{b/n} \operatorname {Gamma}\left (\frac {1-b}{n}\right )+c_1 (-1)^{\frac {1}{n}} n\right )}{n} \]