The 4th order ODE is
$$(D^2-k^2)^2f=0, \qquad (1)$$
where $f=f(y)$, $D\equiv\frac{d}{dy}$, and $k$ is a constant. It is subject to the boundary conditions $f(0)=f'(0)=f(1)=0$.
A solution to (1) is given by,
$$f=a\left(\sinh k\eta+\frac{k\cosh k-\sinh k}{\sinh k}\eta\sinh k\eta-k\eta\cosh k\eta\right), \qquad (2)$$
where $a$ is constant, $\eta=\frac{y}{d}$ with $d$ a constant.
If I consider its characteristic equation:
$$r^4-2r^2k^2+k^4=0,$$
its solutions are $r_{1,2}=r_{3,4}=\pm k$, they then given a general solution
$$f=(\sinh ky+\cosh ky)(c_1+c_2y)+(\cosh ky-\sinh ky)(c_3+c_4y)$$
But I can not figure out how to obtain the solution (2)?
Any suggestion? Thank you!