Internal problem ID [10999]
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: 165.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {\left (a^{2}+x^{2}\right ) y^{\prime \prime }+2 b x y^{\prime }+b \left (b -1\right ) y=0} \]
✓ Solution by Maple
Time used: 0.046 (sec). Leaf size: 33
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)
\[ y \left (x \right ) = c_{1} \left (-i x +a \right )^{-b +1}+c_{2} \left (i x +a \right )^{-b +1} \]
✓ Solution by Mathematica
Time used: 0.813 (sec). Leaf size: 101
DSolve[(x^2+a^2)*y''[x]+2*b*x*y'[x]+b*(b-1)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {\left (a^2+x^2\right )^{\frac {1}{2}-\frac {b}{2}} e^{-i \sqrt {(b-1)^2} \arctan \left (\frac {a}{x}\right )} \left (i c_2 e^{2 i \sqrt {(b-1)^2} \arctan \left (\frac {a}{x}\right )}+2 a \sqrt {(b-1)^2} c_1\right )}{2 a \sqrt {(b-1)^2}} \]