Internal problem ID [10892]
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: 68.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {x y^{\prime \prime }+a y^{\prime }+b \,x^{n} \left (-b \,x^{n +1}+a +n \right ) y=0} \]
✓ Solution by Maple
Time used: 0.594 (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)
\[ y \left (x \right ) = -\left (\left (a -n -2\right ) x^{-\frac {3 n}{2}-\frac {a}{2}-1}+2 b \,x^{-\frac {n}{2}-\frac {a}{2}}\right ) c_{2} \left (n +1\right ) \operatorname {WhittakerM}\left (\frac {-a -n}{2 n +2}, \frac {-a +2 n +3}{2 n +2}, -\frac {2 b \,x^{n +1}}{n +1}\right )+x^{-\frac {3 n}{2}-\frac {a}{2}-1} c_{2} \left (a -n -2\right )^{2} \operatorname {WhittakerM}\left (\frac {n +2-a}{2 n +2}, \frac {-a +2 n +3}{2 n +2}, -\frac {2 b \,x^{n +1}}{n +1}\right )+c_{1} {\mathrm e}^{-\frac {b \,x^{n +1}}{n +1}} \]
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[x*y''[x]+a*y'[x]+b*x^n*(-b*x^(n+1)+a+n)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
Not solved