I guess the meaning of making this calculation is to get a grasp on the effect os order partial derivation on your periodical function.So is a fifth order PDiff, order 3 in y and 2 in x of cos(x*Y).
I guess you have 6 ways to get to the PDiff requested.
The question might be: "which one I choose?"
"I first derive twice for x then three times for y, or the opposite?"
Or, " I alternate the derivation firs by y, then by x, ... by y ...?"
Just write the formula for your equation. Personally, in order to haveexplicitly the order of derivation on the left side I do not define theequation, but I equate f(x,y)=cos(x*y). Then I ask Maple to do the job,I simply select the equation and choose differentiate for x then for yor viceversa up to the fifth degree twice by x and three times by y. Idid ite for the different possible combination in precedence of x and y.
The result is that no matter the derivation pattern already at thefourth differentiation cicle the result is the same for all thecombination in sequence.
Peraphs is this the purpose of you theacher, making you aware of howdifferentiation looses memory of the specificity of differential orderafter the thirt l cicle (3°) in the particular case od cos(x,y).
By substituting the general f(x,y) partial derivatives, in place ofsin(x*y) and cos(x*y) you'll obtain a fourth or fifth degree PDE, whichdoesn't know anymore from were it cames from... A triky bit going backto cos(x*y) from there, assuming yourself had an amnesia .. eheh. Plus in doing this substitution you'll have 2 different possibilities having to use the prime partial derivatives.
I wish I could put a formula here, but though I'd followed the instructions, just a x appeared. Equ1 and Equ2, the fifth order Pdiff and one PDE by substitution are among my files oon the Maple account, hope you could see them from there. Nevetheless you find enclosed the two HTLM in the PS
I might be wrong, I'm not a mathematician, this is all I can think theexercise mean to do. Check it with other friends on Maple Primes, your teacher himself.
<a href='http://www.mapleprimes.com/viewfile/2490'><img src='http://www.mapleprimes.com/scripts/image.php?image=http://www.mapleprimes.com/files/7995_Eqn1.gif&width=300&height=300'/></a></a>
<a href='http://www.mapleprimes.com/viewfile/2491'><img src='http://www.mapleprimes.com/scripts/image.php?image=http://www.mapleprimes.com/files/7995_Eqn2.gif&width=300&height=300'/></a></a>