61.27.34 problem 44

Internal problem ID [12544]
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
Problem number : 44
Date solved : Tuesday, January 28, 2025 at 03:21:12 AM
CAS classification : [[_2nd_order, _missing_y]]

\begin{align*} y^{\prime \prime }+a \,x^{n} y^{\prime }&=0 \end{align*}

✓ Solution by Maple

Time used: 0.002 (sec). Leaf size: 203

dsolve(diff(y(x),x$2)+a*x^n*diff(y(x),x)=0,y(x), singsol=all)
 

\[ y = \frac {x^{-n} \left (c_{2} {\mathrm e}^{-\frac {x^{n} a x}{2 n +2}} \left (\frac {a \,x^{n} x}{n +1}\right )^{\frac {-n -2}{2 n +2}} \left (n +2\right )^{2} \left (n +1\right )^{2} \operatorname {WhittakerM}\left (\frac {n +2}{2 n +2}, \frac {2 n +3}{2 n +2}, \frac {a \,x^{n} x}{n +1}\right )+c_{2} {\mathrm e}^{-\frac {x^{n} a x}{2 n +2}} \left (\frac {a \,x^{n} x}{n +1}\right )^{\frac {-n -2}{2 n +2}} \left (n +1\right )^{3} \left (x^{n} a x +n +2\right ) \operatorname {WhittakerM}\left (-\frac {n}{2 n +2}, \frac {2 n +3}{2 n +2}, \frac {a \,x^{n} x}{n +1}\right )+c_{1} x^{n} \left (n +2\right )\right )}{n +2} \]

✓ Solution by Mathematica

Time used: 0.037 (sec). Leaf size: 56

DSolve[D[y[x],{x,2}]+a*x^n*D[y[x],x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to c_2-\frac {c_1 x \left (\frac {a x^{n+1}}{n+1}\right )^{-\frac {1}{n+1}} \Gamma \left (\frac {1}{n+1},\frac {a x^{n+1}}{n+1}\right )}{n+1} \]