Internal problem ID [10929]
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: 95.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {x y^{\prime \prime }+\left (x^{n} a +b \right ) y^{\prime }+a \left (b -1\right ) x^{-1+n} y=0} \]
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 134
dsolve(x*diff(y(x),x$2)+(a*x^n+b)*diff(y(x),x)+a*(b-1)*x^(n-1)*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = c_{1} x^{-b +1}+c_{2} {\mathrm e}^{-\frac {a \,x^{n}}{2 n}} \left (n \left (\left (n +b -1\right ) x^{\frac {1}{2}-\frac {3 n}{2}-\frac {b}{2}}+a \,x^{\frac {1}{2}-\frac {b}{2}-\frac {n}{2}}\right ) \operatorname {WhittakerM}\left (\frac {b -n -1}{2 n}, \frac {2 n +b -1}{2 n}, \frac {a \,x^{n}}{n}\right )+\operatorname {WhittakerM}\left (\frac {n +b -1}{2 n}, \frac {2 n +b -1}{2 n}, \frac {a \,x^{n}}{n}\right ) x^{\frac {1}{2}-\frac {3 n}{2}-\frac {b}{2}} \left (n +b -1\right )^{2}\right ) \]
✓ Solution by Mathematica
Time used: 0.182 (sec). Leaf size: 90
DSolve[x*y''[x]+(a*x^n+b)*y'[x]+a*(b-1)*x^(n-1)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to (-1)^{-\frac {b}{n}} n^{\frac {b-n-1}{n}} a^{\frac {1-b}{n}} \left (x^n\right )^{\frac {1-b}{n}} \left ((b-1) c_1 (-1)^{b/n} \Gamma \left (\frac {b-1}{n},0,\frac {a x^n}{n}\right )+c_2 (-1)^{\frac {1}{n}} n\right ) \]