# Determine coefficients of a differential equation

1. Nov 20, 2015

### jackonelli

Hi there.

I've been struggling with this problem for days now (4 days, no joke) and I feel like I have a mental block and really cannot get any further.

I have a system that's described by

$$f(t) = g''(t) + 15g'(t) + 1600g(t)$$ Where the input is $$g(t)$$
The problem is to, with this information, determine the coefficients in another system, where the input is$$f(t)$$and the output is given by $$u(t)$$, so that $$u(t) = g(t)$$
This other system is given by
$$c_2u''(t)+c_1u'(t)+c_0u(t) = b_2f''(t) + b_1f'(t) + b_0f(t)$$

I think this is supposed to be simple and I think I make it more difficult in my head than it is. I first substituted f(t) in the second differential equation with the left hand side of the first equation:

$$c_2u''(t)+c_1u'(t)+c_0u(t) = b_2f''(t) + b_1f'(t) + b_0(g''(t) + 15g'(t) + 1600g(t))$$
$$=> c_2u''(t)+c_1u'(t)+c_0u(t) = b_2f''(t) + b_1f'(t) + b_0g''(t) + b_015g'(t) + b_01600g(t)$$

I´m supposed to get numerical values for all coefficients but I really can’t figure out what coefficients makes u(x) = g(t). I don't really know how to proceed. I've tried a lot of other ways too, for example solve for u(t) and g(t) explicitly:

$$u(t) = \frac{b_2}{c_0}f''(t) + \frac{b_1}{c_0}f'(t) + \frac{b_0}{c_0}f(t)-\frac{c_2}{c_0}u''(t)-\frac{c_1}{c_0}u'(t)$$ and $$g(t) = \frac{1}{1600}g''(t) + \frac{15}{1600}g'(t) -\frac{1}{1600}f(t)$$

And then putting their right hand sides equal each other, but this didn't really get me anywhere.

Just before posting this question I’ve been starring and trying for another 4 hours, and I feel I’m that my self-confidence is at an all-time low and I'm starting to ask myself if I really should be doing math at all (Yes, it's really a first world issue, I know).

REALLY grateful for any help!

2. Nov 20, 2015

### jackonelli

Ok, sorry. Actually, putting the last two equations equal to each other really does give values to the coefficients. So I think that I managed to solve it after all This was only a sub-problem, so maybe I'll be back later...