@mehdibaghaee I can't tell whether it is intentional or not, but each of the 5 differential equations you have shown contains a single unknown.  Therefore each of the 5 equations may be solved independently of the other others.  In short, the equations are decoupled.

There is no symbolic solution, in general, for second order equations with variable coefficients. In the two concrete cases that you have shown, the coefficients seem to be taken quite arbitrarily, and therefore you should not expect symbolic solutions.  The following code confirms that Maple is unable to find symbolic solutions:

de := (t-sin(t))*diff(x(t),t,t) + (1-t^3)*diff(x(t),t) + (t^2+t-1)*x(t) = cos(t) - t;
dsolve(de, x(t));   # returns unevaluated
de := (t^5-1)*diff(x(t),t,t) + (t-t^3)*diff(x(t),t) + (t^5 + t^2 + 2)*x(t) = t^3 + 1;
dsolve(de, x(t));   # returns unevaluated

Special cases of second order equations that admit symbolic solutions are discussed extensively in most books on differential equations.  Look up, for example, Bessel's equation, Legendre's equations, Euler's equations, etc.

Of course, differential equations with variable coefficients may be solved numerically if you specify explicit expressions for the coefficients, and provide appropriate initial conditions.

Both expressions that you have provided for W are syntactically correct in Maple, so the meaning of "Please tell me how to write such a conditional maple statement" is unclear.  You need to explain more details to get a helpful answer.

I don't know exactly how long it's been, but I know that it's been quite a while.

I hope that this website's maintainer is not out to outdo Rip Van Winkle.

@Suy I don't see a question in what  @tomleslie has written.  I see the question "cartesian ???" in your post but I don't understand what you mean by that.

If you need further input, then edit and add a corrected statement in your original post. Don't delete any of the previous material because that will invalidate the existing answers.  

If I understand your worksheet correctly, you ask Maple to evaluate

and Maple says "undefined", which is the correct answer.  But you have added a comment that says "This isn't a very useful answer and its wrong".  What do you expect to get for the value of that integral?

@Sradharam Your original question said: "Suppose u and v are dependent variables and x,y,z are independent variables".  In what you posted in your reply you have p as the dependent variable and x, y, z, u, v as the independent variables.

You won't get many useful replies by just throwing apparently random equations, without precise statements and without explaining why you are interested in these, the context in which they arise, and what the boundary conditions are.

What you have shown are the ordinary differential equations that arise from solving a system of PDEs, but you have not shown the PDEs themselves.  What are they?

@acer That's excellent. So it turns out that the solution is not as complicated as it initially seemed, but certainly not obvious, at least to me.  I am going to save this for future use.  Vote up!

The worksheet gives the impression that it was typed when you were half-asleep.  Look over the worksheet again and see if you can fix a large number of obvious errors.  Among other things, you need to make up your mind whether you want a time-dependent solution or a static solution.  Your worksheet shows a mix of the two.

But let me tell you right now that Maple is not equipped to solve the time-dependent plate, so for the time-dependent case you are out of luck.

@Carl Love Great!  This one is a keeper.

@Carl Love I have been looking for a way to plot solutions of ODEs with parameters so I was happy to see your clever construction.

However, there seems to be a problem with the "option remember" in the Ndelta proc.  It appears that it lumps all the delta arguments into one.  See the simple example plotting-odes-with-parameters.mw where I am applying your method to solve a much simpler ODE.  Comment out the "option remember" to obtain the correct plot.

I hope that I am making a dumb mistake and that your construction can be used to produce the correct plot.

@pasty When polygonplot3d is called with a matrix as argument, it is expected that the matrix is n×3, where each of the n rows holds the coordinates of a vertex of the polygon.  The documentation says that if the matrix is 3×n and n≠3, then it is automatically transposed into a n×3 matrix.

The case of n=3 (plotting a triangle) requires close attention.  Is it rows or columns that are vertex coordinates?  It is the rows!  That's a consequence of the n≠3 clause noted above.

The moral of the story? Don't rely on authomatic transposition.  Always prrovide vertex coordinates as rows!

@ik74 There is no single source that covers everything that you need to know on this subject. You need to combine knowledge from various areas.  Here is a brief summary.

1.  The product rule for differentiation from calculus.  For any two functions u(x) and v(x) we have
   [u(x) v(x)]' = u'(x) v(x) + u(x) v'(x),
   which we use it in the equivalent form
   u'(x) v(x) = [u(x) v(x)]' - u(x) v'(x),
2. The summation convention (also known as the Einstein notation) for vectors, tensors, and partial derivatives, is introduced in most books on continuum mechanics.  For instance, A First Course in Continuum Mechanics by Oscar Gonzalez and Andrew Stuart.  According to the summation convention, the dot product of the vectors u and v is written as u_i v_i, where a summation over the repeated index i is implicitly assumed. The gradient of the vector u is a second order tensor expressed as u_i,j.. This is the notation I used in my derivation that I posted earlier.
3. The Divergence Theorem is essential for deriving the weak formulation. If your calculus book does not cover the Divergence Theorem, let me know the name of the book and I will recommend an alternative.
4. The weak formulation of PDEs is in covered in advanced books on partial differential equations.  For instance, Partial Differential Equations by Evans.
5. The finite element approximation of the solutions of PDEs is a vast subject. I don't know your educational background so I cannot recommend a good starting point.
6. A good grounding in the theory of elasticity. Since you are referring to the Navier equation, you probably have a reference for that.

Anyway, getting to where you are aiming for is a long (but rewarding) journey.  You should realize that you cannot skip the intermediate steps and jump to the end.  That's why I recommended beginning with an easier problem and then gradually working your way up. Ask one of your teachers for guidance.

@ik74 Here is the calculation of the weak formulation of the problem:

weak-formulation.pdf

As I said before, if you have to ask how to do this, it means that you are not ready to handle such problems yet.  It would be advisable to begin with much more elementary problems and gradually work your way up. This one is quite advanced.

Another thing: Although you said that this is a 3D problem in the space variables, I seriously doubt it.  It looks to me that the PDE is the equation of the vibration of of a 2D plate.

@ik74 Solving that equation in 3D is a nontrivial task.  Is there a reason that you are interested in that PDE in 3D? Have a look at this paper to see where the difficulties lie.

I am concerned that on the one hand you are asking for help with the weak formulation of the PDE which is a trivial task, and on the other hand you are aiming to implement a finite elements algorithm for the PDE in 3D which is a very nontrivial task.  It's not clear to me where your strengths are.  Perhaps you should begin with the simplest possible problem and then gradually advance to more complex ones.