Hi, is there some method for eliminating dual solutions to a given differential equation that are not wanted? Let me explain: Consider the differential equation

$\displaystyle \frac{\partial^2 y}{\partial t^2}=\hat{C}y\ (^*),$

where $\displaystyle \hat{C}$ is a linear operator (that doesn't operate in the time dimension). However, since the equation is going to be solved numerically, and the operator $\displaystyle \hat{C}$ is very computationally expensive to be implemented directly, the equation is modified:

$\displaystyle \frac{\partial^2 y}{\partial t^2}=\hat{C}y$

$\displaystyle \Rightarrow \frac{\partial^2}{\partial t^2}\frac{\partial^2 y}{\partial t^2}=\frac{\partial^2}{\partial t^2}\hat{C}y$

$\displaystyle \Leftrightarrow\frac{\partial^4 y}{\partial t^4}=\hat{C}\frac{\partial^2 y}{\partial t^2}$

$\displaystyle \Leftrightarrow\frac{\partial^4 y}{\partial t^4}=\hat{C}^2y\ (^{**}),$

where $\displaystyle \hat{C}^2$ is much computationally cheaper to implement. However, this equation gives dual solutions that do not satisfy the original equation (*), since (**) can be shown to allow any linear combination of solutions to the equations

$\displaystyle \frac{\partial^2 y}{\partial t^2}=\hat{C}y$ (*, our original equation)

and to

$\displaystyle \frac{\partial^2 y}{\partial t^2}=-\hat{C}y\ (^{***}).$

So the modified equation can't be used solely. Is there some way to eliminate the solutions given by (***) when solving the system numerically, and only obtain solutions given by (*)? Thanks in advance.