Internal problem ID [8959]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1380.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {y^{\prime \prime }+\frac {b y}{x^{2} \left (x -a \right )^{2}}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.007 (sec). Leaf size: 75
dsolve(diff(diff(y(x),x),x) = -b/x^2/(x-a)^2*y(x),y(x), singsol=all)
\[ y \relax (x ) = c_{1} \sqrt {x \left (a -x \right )}\, \left (\frac {a -x}{x}\right )^{\frac {\sqrt {a^{2}-4 b}}{2 a}}+c_{2} \sqrt {x \left (a -x \right )}\, \left (\frac {x}{a -x}\right )^{\frac {\sqrt {a^{2}-4 b}}{2 a}} \]
✓ Solution by Mathematica
Time used: 0.247 (sec). Leaf size: 121
DSolve[y''[x] == -((b*y[x])/(x^2*(-a + x)^2)),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} 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}}} \\ \end{align*}