I am considering a relation of the form $$T\frac{\partial u}{\partial t}+\frac{1}{2}T^2\frac{\partial ^2 u}{\partial t^2} = \left(p_1-L\frac{\partial p_1}{\partial x}\right)\left(u-L\frac{\partial u}{\partial x}+\frac{1}{2}L^2\frac{\partial ^2u}{\partial x^2}\right)+\left(p_2+L\frac{\partial p_2}{\partial x}\right)\left(u+L\frac{\partial u}{\partial x}+\frac{1}{2}L^2\frac{\partial ^2u}{\partial x^2}\right)+\left(p_3-L\frac{\partial p_3}{\partial y}\right)\left(u-L\frac{\partial u}{\partial y}+\frac{1}{2}L^2\frac{\partial ^2u}{\partial y^2}\right)+\left(p_4+L\frac{\partial p_4}{\partial y}\right)\left(u+L\frac{\partial u}{\partial y}+\frac{1}{2}L^2\frac{\partial ^2u}{\partial y^2}\right),$$
subject to $p_1+p_2+p_3+p_4=1.$
I want to do some manipulations on this expression, namely:
1) Take the derivative on both sides w.r.t $t$ and discard third order derivatives or higher. This will give an expression for $\frac{\partial^2 u}{\partial t^2}$ in terms of partial derivatives w.r.t $t$ of the functions on the right hand side.
2) Take the derivative of both sides w.r.t $x$ and $y$ respectively, once again discarding derivatives of order three and higher. This will give expressions for $\frac{\partial^2 u}{\partial x \partial t}$ and $\frac{\partial^2 u}{\partial y \partial t}$ in terms of partial derivatives w.r.t to $x$ and $y$ of the right hand side.
3) Substitute the results from step 2 into step 1 for the expression of $\frac{\partial^2 u}{\partial t^2}$.
4) Take the expression for $\frac{\partial^2 u}{\partial t^2}$ from step 3 and substitute into the original equation. Collect the terms in terms of $u$ and partial derivatives of $u$.
This gets very messy if one tries to do it by hand, so I was thinking about using a CAS such as Maple. I recently asked a related question on SO about this.
If somebody could give me some hints on how one might do these manipulations in Maple, I would be very grateful.