Internal problem ID [11050]
Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev.
Second edition
Section: Chapter 2, Second-Order Differential Equations. section 2.1.2-7 Equation of form
\((a_4 x^4+a_3 x^3+a_2 x^2 x+a_1 x+a_0) y''+f(x)y'+g(x)y=0\)
Problem number: 216.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {x^{2} \left (x -a \right )^{2} y^{\prime \prime }+b y=0} \]
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 75
dsolve(x^2*(x-a)^2*diff(y(x),x$2)+b*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = c_{1} \sqrt {x \left (-x +a \right )}\, \left (\frac {-x +a}{x}\right )^{\frac {\sqrt {a^{2}-4 b}}{2 a}}+c_{2} \sqrt {x \left (-x +a \right )}\, \left (\frac {x}{-x +a}\right )^{\frac {\sqrt {a^{2}-4 b}}{2 a}} \]
✓ Solution by Mathematica
Time used: 0.487 (sec). Leaf size: 121
DSolve[x^2*(x-a)^2*y''[x]+b*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {x^{\frac {1}{2}-\frac {1}{2} \sqrt {1-\frac {4 b}{a^2}}} (x-a)^{\frac {1}{2}-\frac {1}{2} \sqrt {1-\frac {4 b}{a^2}}} \left (a c_1 \sqrt {1-\frac {4 b}{a^2}} x^{\sqrt {1-\frac {4 b}{a^2}}}+c_2 (x-a)^{\sqrt {1-\frac {4 b}{a^2}}}\right )}{a \sqrt {1-\frac {4 b}{a^2}}} \]