Find maximum of summation of functions

[dimensionless]

$$y(t) = \sum C_{n} e^{-\gamma t} sin(n \omega t)$$

y(t) is a summation of a large number of arbitrarily decaying sinusoids with arbitrary coefficients. Find the value of t at which y(t) is a maximum.



Personally, I have doubts that this can be solved without knowing what the constants and are and what gamma is. This is not a homework problem, so I do not know for a fact that it is solvable. What I want is some equation for t. The problem is that there could be a large number of minimums and maximums. Does anyone think this is possible without using a lot of logic and elbow grease?

 

[mighty2000]

I understand your concerns about solving this problem without knowing all the necessary information. However, there are some general principles and techniques that can be applied to find the value of t at which y(t) is a maximum.

Firstly, we can look at the general shape of the function y(t). Since it is a summation of decaying sinusoids with arbitrary coefficients, we can expect the function to have a periodic behavior with a decreasing amplitude over time. This means that there will be multiple maximums and minimums throughout the function.

To find the value of t at which y(t) is a maximum, we can use techniques from calculus. The first step would be to take the derivative of y(t) with respect to t. This will give us an expression that represents the rate of change of y(t) at any given time t.

Next, we can set this derivative equal to zero and solve for t. This will give us the critical points, where the function either reaches a maximum or minimum. We can then use the second derivative test to determine whether these points are maximums or minimums.

If we find that the critical points are maximums, we can then plug these values of t back into the original function y(t) to find the corresponding maximum value of y(t).

It is important to note that this approach may not give us a specific value of t but rather a range of values where y(t) is a maximum. This is due to the fact that there could be multiple maximums in the function, as mentioned earlier.

In conclusion, while it may not be possible to find a specific value of t at which y(t) is a maximum without knowing all the necessary information, we can use techniques from calculus to find a range of values where this occurs. This approach may require some trial and error and may not give a definitive answer, but it can still provide valuable insights into the behavior of the function.