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Mathematica (diffeq) - solving a function inside of a diffeq in terms of another function

Hello,  I've been scratching my head on a particular question for hours now, and it's pretty clear that I need help.  I have a diffeq:

y'[t]==d[t] (40-y[t])+0.5 (o[t]-y[t])
y[0]==starter



, and I am supposed to come up with a function d[t] expressed in terms of o[t] so that the solutions of y[t] of the function diffeq above settle in at about 50.  How would I go about doing this?  I've tried separation of variables and DSolve to no avail.  Any ideas?  I need a good explanation more than just the answer.  Thanks!  o[t], incidentally, equals 80-11 Cos[[Pi t)/12]+3.25691 Sin[Pi t/12]
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dilligent
Asked:
dilligent
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phoffricCommented:
define o(t) = 80 + S(t), where S(t) is a sinusoidal function = -11 Cos[[Pi t)/12]+3.25691 Sin[Pi t/12]
   -- well your parenthesis are not balanced, so it isn't clear what the exact function is supposed to be, but it doesn't matter. S(t) is sinusoidal with some min/max range.

>> settle in at about 50
Assume you mean as t -> oo
At t = oo, y = 50
At t = oo, dy/dt = ?

Plug these two values into your DE at t = oo, and express in terms of S(t) and d(t). Then solve for d(t) = f( S(t) ). This should remove the oscillation.
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dilligentAuthor Commented:
Sorry, this actually was related to what the expert posted.  I need to credit him.
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dilligentAuthor Commented:
Not sure how to cancel the close request.
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phoffricCommented:
Would you like to post the steps to your solution for review. I think I got a different answer than you (but maybe not, since your answer uses different symbols than in your OP). For example, you final answer began with c[t_]= ...
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