Internal problem ID [10868]
Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev.
Second edition
Section: Chapter 2, Second-Order Differential Equations. section 2.1.2-2 Equation of form
\(y''+f(x)y'+g(x)y=0\)
Problem number: 44.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _missing_y]]
\[ \boxed {y^{\prime \prime }+a \,x^{n} y^{\prime }=0} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 244
dsolve(diff(y(x),x$2)+a*x^n*diff(y(x),x)=0,y(x), singsol=all)
\[ y \left (x \right ) = \frac {x^{-n} \left (\left (\frac {a x \,x^{n}}{n +1}\right )^{\frac {-n -2}{2 n +2}} c_{2} \left (\frac {a}{n +1}\right )^{\frac {1}{n +1}} {\mathrm e}^{-\frac {x^{n} a x}{2 n +2}} \left (n +2\right )^{2} \left (n +1\right )^{2} \operatorname {WhittakerM}\left (\frac {n +2}{2 n +2}, \frac {2 n +3}{2 n +2}, \frac {a x \,x^{n}}{n +1}\right )+\left (\frac {a x \,x^{n}}{n +1}\right )^{\frac {-n -2}{2 n +2}} c_{2} \left (\frac {a}{n +1}\right )^{\frac {1}{n +1}} {\mathrm e}^{-\frac {x^{n} a x}{2 n +2}} \left (n +1\right )^{3} \left (x^{n} a x +n +2\right ) \operatorname {WhittakerM}\left (-\frac {n}{2 n +2}, \frac {2 n +3}{2 n +2}, \frac {a x \,x^{n}}{n +1}\right )+2 c_{1} \left (n +2\right ) a \left (n +\frac {3}{2}\right ) x^{n}\right )}{\left (n +2\right ) \left (2 n +3\right ) a} \]
✓ Solution by Mathematica
Time used: 0.054 (sec). Leaf size: 56
DSolve[y''[x]+a*x^n*y'[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to c_2-\frac {c_1 x \left (\frac {a x^{n+1}}{n+1}\right )^{-\frac {1}{n+1}} \Gamma \left (\frac {1}{n+1},\frac {a x^{n+1}}{n+1}\right )}{n+1} \]