Internal problem ID [8961]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1382.
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 {c y}{\left (x -a \right )^{2} \left (x -b \right )^{2}}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 116
dsolve(diff(diff(y(x),x),x) = c/(x-a)^2/(x-b)^2*y(x),y(x), singsol=all)
\[ y \relax (x ) = c_{1} \sqrt {\left (-x +a \right ) \left (b -x \right )}\, \left (\frac {-x +a}{b -x}\right )^{\frac {\sqrt {a^{2}-2 b a +b^{2}+4 c}}{2 a -2 b}}+c_{2} \sqrt {\left (-x +a \right ) \left (b -x \right )}\, \left (\frac {-x +a}{b -x}\right )^{-\frac {\sqrt {a^{2}-2 b a +b^{2}+4 c}}{2 a -2 b}} \]
✓ Solution by Mathematica
Time used: 0.639 (sec). Leaf size: 141
DSolve[y''[x] == (c*y[x])/((-a + x)^2*(-b + x)^2),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to (x-a)^{\frac {1}{2} \left (1-\sqrt {\frac {4 c}{(a-b)^2}+1}\right )} (x-b)^{\frac {1}{2} \left (1-\sqrt {\frac {4 c}{(a-b)^2}+1}\right )} \left (c_1 (x-a)^{\sqrt {\frac {4 c}{(a-b)^2}+1}}-\frac {c_2 (x-b)^{\sqrt {\frac {4 c}{(a-b)^2}+1}}}{(a-b) \sqrt {\frac {4 c}{(a-b)^2}+1}}\right ) \\ \end{align*}