I have a 2nd order ODE of the form:
$\ddot {x} + 2c \dot {x} + 39Ex = 0 $
$Initial$ conditions being: x(0) = 0 and $\dot {x}(0)$=0.1
Where c is in the interval [1,5] and E is in the interval [3,15]
Note that x is a function of time x=> x(t)
One needs to find value for c and E, such that the absolute of ${x}< 0.002$ and absolute of : $\dot {x} < 0.02 $ (over 3 cycles of the solution to the ODE)
So the way I see this is that one has to go thru say many values of c and E, to ensure that x and its derivative satisfy the bounds. So that means that this really is just a constant coefficient ODE. So I assume that c and E are some constants that will be plugged into the equation and then the ODE would be solved(like in Matlab) and one can check the time interval over which the solutions satisfies this condition.
So I solved this ODE with the initial conditions which gives: $ x(t) = 0.1/z * exp(-ct)*sin(zt)$ where $z= \sqrt{39E-c}$
So my question is this: Do I have to resort to using a brute force way of choosing values of c and E within their particular intervals.(Note I am using Matlab to aid in solving this problem). OR is there a better approach. I was thinking that maybe one can setup some sort of system of nonlinear algebraic equations and somehow solve these.(Once again use of Matlab is OK).
I hope someone can give me input into how to approach this problem.
Regards,
P