Internal problem ID [10926]
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: 92.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {x y^{\prime \prime }+\left (x^{n} a +2\right ) y^{\prime }+x^{-1+n} y a=0} \]
✓ Solution by Maple
Time used: 0.032 (sec). Leaf size: 105
dsolve(x*diff(y(x),x$2)+(a*x^n+2)*diff(y(x),x)+a*x^(n-1)*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = \frac {c_{1}}{x}+c_{2} \left (\left (\left (n +1\right ) x^{-\frac {3 n}{2}-\frac {1}{2}}+a \,x^{-\frac {n}{2}-\frac {1}{2}}\right ) n \operatorname {WhittakerM}\left (-\frac {1}{2}+\frac {1}{2 n}, 1+\frac {1}{2 n}, \frac {a \,x^{n}}{n}\right )+x^{-\frac {3 n}{2}-\frac {1}{2}} \operatorname {WhittakerM}\left (\frac {1}{2}+\frac {1}{2 n}, 1+\frac {1}{2 n}, \frac {a \,x^{n}}{n}\right ) \left (n +1\right )^{2}\right ) {\mathrm e}^{-\frac {a \,x^{n}}{2 n}} \]
✓ Solution by Mathematica
Time used: 0.099 (sec). Leaf size: 62
DSolve[x*y''[x]+(a*x^n+2)*y'[x]+a*x^(n-1)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to (-1)^{-1/n} n^{\frac {1}{n}-1} a^{-1/n} \left (x^n\right )^{-1/n} \left (c_1 (-1)^{\frac {1}{n}} \Gamma \left (\frac {1}{n},0,\frac {a x^n}{n}\right )+c_2 n\right ) \]