anonymous
  • anonymous
Mostly solved, just help on one step! Find the general solution and use it to determine how solutions behave as t->inf: y'+(1/t)y=3cos(2t), t>0
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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jamiebookeater
  • jamiebookeater
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inkyvoyd
  • inkyvoyd
o.o first order ODE :o
anonymous
  • anonymous
I found the solution as: \[y=(3/2)\sin(2t)+(3/4t)\cos(2t)+(c/t)\] which it says is correct, but then it goes on to say that the general solution is: \[y=(3/2)\sin(2t)\] and I'm not sure how they got that
anonymous
  • anonymous
I think it has something to do with the solution changing as t->infinity

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inkyvoyd
  • inkyvoyd
as t approacs infinity the other two expressions approach zero.
inkyvoyd
  • inkyvoyd
Is it (3/(4t))cos(2t) or (3t/4)cos(2t)?
anonymous
  • anonymous
It is (3/(4t))cos(2t)
inkyvoyd
  • inkyvoyd
\(\large \lim_{t->\inf} (3/2)\sin(2t)+\frac{3}{4t}\cos(2t)+(c/t)\) \(=(3/2)\sin(2t)+0*\cos(2t)+0\)
inkyvoyd
  • inkyvoyd
essentially as t approaches infinity we still see oscillating behavior in y, but the effects of the cos term and c/t are greatly reduced to the point they are negligible.
anonymous
  • anonymous
Thank you! If I just take the limit as t->infinity, then what is the point of adding that t>0?
inkyvoyd
  • inkyvoyd
I'm not particularly sure either, but I will try to make sense out of it.
anonymous
  • anonymous
Alright, well thank you!
inkyvoyd
  • inkyvoyd
There may be a case where you take the natural logarithm of an expression, and in that case t must be greater than 0. I think that is why they have that restriction.
inkyvoyd
  • inkyvoyd
I mean while doing the computation, you must take the natural logarithm with an expression with t or a multiple of t, and if t<=0 the operation is undefined.
anonymous
  • anonymous
thank you!
inkyvoyd
  • inkyvoyd
you're welcome :)

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