Internal problem ID [10972]
Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev.
Second edition
Section: Chapter 2, Second-Order Differential Equations. section 2.1.2-4 Equation of form
\(x^2 y''+f(x)y'+g(x)y=0\)
Problem number: 138.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {x^{2} y^{\prime \prime }+\left (a \,x^{2}+b x \right ) y^{\prime }+\left (k \left (a -k \right ) x^{2}+\left (a n +b k -2 k n \right ) x +n \left (-n +b -1\right )\right ) y=0} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 48
dsolve(x^2*diff(y(x),x$2)+(a*x^2+b*x)*diff(y(x),x)+(k*(a-k)*x^2+(a*n+b*k-2*k*n)*x+n*(b-n-1))*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = c_{1} x^{-n} {\mathrm e}^{-k x}+c_{2} \operatorname {WhittakerM}\left (-\frac {b}{2}+n , -\frac {b}{2}+n +\frac {1}{2}, \left (-2 k +a \right ) x \right ) x^{-\frac {b}{2}} {\mathrm e}^{-\frac {a x}{2}} \]
✓ Solution by Mathematica
Time used: 0.504 (sec). Leaf size: 64
DSolve[x^2*y''[x]+(a*x^2+b*x)*y'[x]+(k*(a-k)*x^2+(a*n+b*k-2*k*n)*x+n*(b-n-1))*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to e^{-k x} x^{-n} \left (c_1-c_2 x^{-b+2 n+1} (x (a-2 k))^{b-2 n-1} \Gamma (-b+2 n+1,(a-2 k) x)\right ) \]