61.28.8 problem 68

Internal problem ID [12568]
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
Problem number : 68
Date solved : Tuesday, January 28, 2025 at 08:02:22 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x y^{\prime \prime }+a y^{\prime }+b \,x^{n} \left (-b \,x^{n +1}+a +n \right ) y&=0 \end{align*}

✓ Solution by Maple

Time used: 2.178 (sec). Leaf size: 166

dsolve(x*diff(y(x),x$2)+a*diff(y(x),x)+b*x^n*(-b*x^(n+1)+a+n)*y(x)=0,y(x), singsol=all)
 

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

✗ Solution by Mathematica

Time used: 0.000 (sec). Leaf size: 0

DSolve[x*D[y[x],{x,2}]+a*D[y[x],x]+b*x^n*(-b*x^(n+1)+a+n)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

Not solved