@Markiyan Hirnyk , nice!  Along with the generic solve method, and the eliminate method, I now have 3 different sets of 4 long solutions (interesting that they all appear different).  I am going to try coding them into Matlab for direct application to my data; it's too bad Matlab doesn't employ a Maple symbolic toolbox, it'll be tricky to avoid typos!  Thanks again for all your effort.

To clarify, given

x₁ := M*(cos(φ)-a*cos(φ)) / (1-b*cos(2*φ))
x₂ := M*(cos(φ)+a*sin(φ)) / (1+b*sin(2*φ))
y₁ := M*(sin(φ)+a*sin(φ)) / (1-b*cos(2*φ))
y₂ := M*(sin(φ)+a*cos(φ))/ (1+b*sin(2*φ))

I'm searching for a closed-form symbolic expression M(x₁,x₂,y₁,y₂)

M, a, b, φ, x₁, x₂, y₁, and y₂ are all real.
M > 0
0<a<1
0<b<1 

@Markiyan Hirnyk, my desire is to find a closed-form symbolic expression for M in terms of x1,x2,y1 and y2. If it isn't possible, so be it, but I'd like to know for sure. 

Regardless, I will test John May's suggestion for a few sets, as I am losing hope!  Thanks...

@Markiyan Hirnyk, my desire is to find a closed-form symbolic expression for M in terms of x1,x2,y1 and y2. If it isn't possible, so be it, but I'd like to know for sure. 

Regardless, I will test John May's suggestion for a few sets, as I am losing hope!  Thanks...

Thanks Markiyan

But if I am going to employ a numerical method, I'll use Matlab, as that is where my data resides.

I was hoping to find a closed form solution to this problem.

Thanks Markiyan

But if I am going to employ a numerical method, I'll use Matlab, as that is where my data resides.

I was hoping to find a closed form solution to this problem.

Thanks, installing Maple 14.01 alleviates the error.  A quick "combine(%, trig)" after sol2 also gives the nice compact form: arctan(sin(w*t)+exp(-r/d)*sin(-w*t+w*r), cos(w*t)-exp(-r/d)*cos(-w*t+w*r))

But this can be obtained by hand with a few lines of simple math; it's too bad it can't apply the argument/arctan function...

Thanks, installing Maple 14.01 alleviates the error.  A quick "combine(%, trig)" after sol2 also gives the nice compact form: arctan(sin(w*t)+exp(-r/d)*sin(-w*t+w*r), cos(w*t)-exp(-r/d)*cos(-w*t+w*r))

But this can be obtained by hand with a few lines of simple math; it's too bad it can't apply the argument/arctan function...

thanks for your time;

I do not have numeric values; t is the independent variable, and r,d, and w are parameters.  I will ultimately use Matlab to run some simulations with numeric parameter values, but here I am simply trying to find a closed-form solution from the information I've given.

I tried your suggestion, but the "assuming real, r>0, d>0, t>0" portions yield errors, and nothing interesting results when a subset of the the assumptions are employed.

Any suggestions?

BTW, the final term can be omitted (it does not afffect the phase):

thanks for your time;

I do not have numeric values; t is the independent variable, and r,d, and w are parameters.  I will ultimately use Matlab to run some simulations with numeric parameter values, but here I am simply trying to find a closed-form solution from the information I've given.

I tried your suggestion, but the "assuming real, r>0, d>0, t>0" portions yield errors, and nothing interesting results when a subset of the the assumptions are employed.

Any suggestions?

BTW, the final term can be omitted (it does not afffect the phase):

I recently discovered an error is my original "I" equations. The Groebner basis methods applied to the new system worked fabulously; solving the quadratic in M gives a tight solution. I worked it out by hand, and confirmed it's validity! Yahoo! Unfortunately I don't understand the Groebner technique or even Maple that well, but your help has given me the final form I needed to confirm my results. Thanks, Robert!

I recently discovered an error is my original "I" equations. The Groebner basis methods applied to the new system worked fabulously; solving the quadratic in M gives a tight solution. I worked it out by hand, and confirmed it's validity! Yahoo! Unfortunately I don't understand the Groebner technique or even Maple that well, but your help has given me the final form I needed to confirm my results. Thanks, Robert!

Bummer. That's too bad, since I know there is a non-zero solution to this problem. I'll keep plugging...

Bummer. That's too bad, since I know there is a non-zero solution to this problem. I'll keep plugging...

Perfect! Thanks for the tips, Robert. Didn't even need simplify! convert(%,trig) and combine(%,trig) did the trick. Separate problem...do you (or anyone else) know how to solve for M (I1,I2,I3,I4) using I1 := M*(1-a*exp(-I*theta))/(1-b*cos(theta)) I2 := M*(1-a*I*exp(I*theta))/(1+b*sin(theta)) I3 := M*(1+a*exp(-I*theta))/(1+b*cos(theta)) I4 := M*(1+a*I*exp(I*theta))/(1-b*sin(theta)) 4 knowns: I1,I2,I3, and I4, 4 unknowns: M, a, b, and theta I want to get to solve for M in terms of I1,I2,I3, and I4 (eliminating a, b, and theta) Mike