Internal problem ID [9027]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1448.
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^{2} y}{\left (-a^{2}+x^{2}\right )^{2}}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.009 (sec). Leaf size: 87
dsolve(diff(diff(y(x),x),x) = -b^2/(-a^2+x^2)^2*y(x),y(x), singsol=all)
\[ y \relax (x ) = c_{1} \sqrt {\left (a -x \right ) \left (x +a \right )}\, \left (\frac {a -x}{x +a}\right )^{\frac {\sqrt {a^{2}-b^{2}}}{2 a}}+c_{2} \sqrt {\left (a -x \right ) \left (x +a \right )}\, \left (\frac {a -x}{x +a}\right )^{-\frac {\sqrt {a^{2}-b^{2}}}{2 a}} \]
✓ Solution by Mathematica
Time used: 0.275 (sec). Leaf size: 127
DSolve[y''[x] == -((b^2*y[x])/(-a^2 + x^2)^2),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {1}{2} (x-a)^{\frac {1}{2}-\frac {1}{2} \sqrt {1-\frac {b^2}{a^2}}} (a+x)^{\frac {1}{2}-\frac {1}{2} \sqrt {1-\frac {b^2}{a^2}}} \left (2 c_1 (x-a)^{\sqrt {1-\frac {b^2}{a^2}}}-\frac {c_2 (a+x)^{\sqrt {1-\frac {b^2}{a^2}}}}{a \sqrt {1-\frac {b^2}{a^2}}}\right ) \\ \end{align*}