# Spectrum of eigen values for coupled differential equations

How do I obtain a spectrum of eigenvalues for my system of coupled differential equations?

$$kf''(\theta) + \epsilon_{1} f(\theta) + a\cos(b \theta + c) g(\theta) = \lambda f(\theta),\\ a\cos(b \theta + c) f(\theta) + k g''(\theta) + \epsilon_{2}g(\theta) = \lambda g(\theta)$$

The system has periodic boundary conditions:

$$f(\theta +2\pi) = f(\theta)\\ g(\theta +2\pi) = g(\theta)$$

I have tried using NDEigensystem, but I would like to plot a spectrum of the eigen values with varying parameters $$\left[ k, \epsilon_{1},\epsilon_{2},a,b,c \right]$$. Here is a bit of code that I tried, but it doesn't work the way I want.

eqns= {k f''[\[Theta]] + Subscript[\[Epsilon], 1 ] f[\[Theta]] + a Cos[b \[Theta] + c] g[\[Theta]],
k g''[\[Theta]] + Subscript[\[Epsilon], 2 ] g[\[Theta]] + a Cos[b \[Theta] + c] f[\[Theta]] }
bc =  PeriodicBoundaryCondition[g[\[Theta]], \[Theta] == 2 \[Pi],Function[\[Theta], \[Theta] - 2 \[Pi]]]
{vals,funs}= NDEigensystem[{eqns,bc}/.{k->1,Subscript[\[Epsilon], 1 ]->1,Subscript[\[Epsilon], 2 ]->1,a->1,b->1,c->1},{f[\[Theta]], g[\[Theta]]}, {\[Theta], 0, 2\[Pi]}, 50]


I am looking for a way to vary the variable I previously mentioned and see how they change using a function similar to Manipulate.

• Would be a good idea to give some Mathematica code for the equations. – KraZug Nov 26 '18 at 9:43
• @VijayMocherla Can you add a minimal working code example of what you tried in your question? This makes it much easier for people to see what you were trying to do and give you a helpful answer. – Thies Heidecke Nov 26 '18 at 19:19
• @ThiesHeidecke Thanks for the suggestion. I have just added that to my original question. – EverydayFoolish Nov 27 '18 at 14:16

Manipulate[