Internal problem ID [10998]
Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev.
Second edition
Section: Chapter 2, Second-Order Differential Equations. section 2.1.2-5 Equation of form
\((a x^2+b x+c) y''+f(x)y'+g(x)y=0\)
Problem number: 164.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {\left (-a^{2}+x^{2}\right ) y^{\prime \prime }+2 b x y^{\prime }+b \left (-1+b \right ) y=0} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 29
dsolve((x^2-a^2)*diff(y(x),x$2)+2*b*x*diff(y(x),x)+b*(b-1)*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = c_{1} \left (x +a \right )^{-b +1}+c_{2} \left (a -x \right )^{-b +1} \]
✓ Solution by Mathematica
Time used: 0.727 (sec). Leaf size: 127
DSolve[(x^2-a^2)*y''[x]+2*b*x*y'[x]+b*(b-1)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {(x-a)^{\frac {1}{2}-\frac {1}{2} \sqrt {(b-1)^2}} (a+x)^{\frac {1}{2}-\frac {1}{2} \sqrt {(b-1)^2}} \left (x^2-a^2\right )^{-b/2} \left (2 a \sqrt {(b-1)^2} c_1 (x-a)^{\sqrt {(b-1)^2}}-c_2 (a+x)^{\sqrt {(b-1)^2}}\right )}{2 a \sqrt {(b-1)^2}} \]