How Can I Modify My Equation to Become a Continuous-Time Function?

[sooyewguan]

Lets say I have an equation,

y=\alpha e^{\beta W}

where,
\alpha = a e^{b f} and \beta = c f + d

W = \int^{T}_{0}f dt

My problem now is, what happen if f is changing with time t, f(t)

How do I modify my main equation, y, so that it become an continuous-time function, y(t).

Thank you.

 

[HallsofIvy]

I'm not sure what you mean: what you give
y= \alpha e^\beta W(t)[/itex]<br /> <b>is</b> a &quot;continuous-time function&quot;- or at least a continuous function of t.<br /> <br /> If you want to you can replace each of \alpha, \beta, and W with their explicit dependence on t:<br /> y(t)= ae^{bf(t)} e^{(cf+d)\int_0^t f(u)du}<br /> (I&#039;ve changed the dummy variable in the integral to u so as not to confuse it with the variable t.)<br /> <br /> But I don&#039;t think that really adds anything as long as you don&#039;t know the explicit form of f.