It is known from Mechanics of Structures that the motion of a beam can be modelled by means of the following partial differential equation:
$EIw^{iv}+\rho{A}\ddot{w} = 0$,
where $EI$ and $\rho{A}$ are constants, and where the following variable transformation can be done:
$w(x,t) = \phi(x)\sin{\omega{t}}$.
Then, the PDE turns into an ODE:
$\phi^{iv}-\beta^4\phi = 0$,
whose solution is well known:
$\phi(x) = C_1\sin{\beta{x}}+C_2\cos{\beta{x}}+C_3\sinh{\beta{x}}+C_4\cosh{\beta{x}}$.
Now, I am trying to study the motion of a beam convered by piezoelectric tapes. In this case, the PDE is
$EIw^{iv}+\rho{A}\ddot{w}+{\alpha}v'' = 0$,
in which $\alpha$ is constant, and the voltage $v(x,t)$ can be written as
$v(x,t) = \hat{v}(t)\left[H(x)-H(x-\ell)\right]$,
where $H(x)$ is the Heaveside function.
As you can observe, for this second PDE an analytical general solution is not straightforward as for the first PDE. Therefore, my question is: will be there any ways ($\textit{e.g.}$: a variable transformation) which will allow me to obtain a closed-form solution for that equation?
Thanks in advance.