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September 12th, 2013, 06:54 PM   #1
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Diff Eq - Solve the initial value problem

,

Let be the value of for which the transition from one type of behavior to another occurs. Find the critical value exactly.

I've transformed the equation into so,

and

I multiply by integrating factor:



Rewrite as:



Then:







and

I feel like I've made this too complicated to find
Can someone give me some tips
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September 12th, 2013, 07:44 PM   #2
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Re: Diff Eq - Solve the initial value problem

Using essentially the same method, I get:

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September 12th, 2013, 09:39 PM   #3
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Re: Diff Eq - Solve the initial value problem

Quote:
Originally Posted by MarkFL
Using essentially the same method, I get:

How were you able to make it look so... neat?

I got the question correct by guessing, but I'm still uncertain on how I'm supposed to find the critical values. I'm assuming you use the first derivative test, but I can't seem to find a zero or an undefined point.
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September 12th, 2013, 10:25 PM   #4
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Re: Diff Eq - Solve the initial value problem

This is my working:

Like you I wrote the ODE in the form:

where

We find the integrating factor is:

and so the ODE becomes:



Writing the left side as the differentiation of a product, we have:



Integrating with respect to , we find:



Multiply through by :



Now we may using the initial value to determine the value of the parameter :



And so the solution satisfying the IVP is:

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September 13th, 2013, 09:24 AM   #5
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Re: Diff Eq - Solve the initial value problem

Quote:
Originally Posted by MarkFL

And so the solution satisfying the IVP is:

I'm still uncertain on how to find

The hint that I'm given is this:

Find the general solution y(t) and solve for y'(0) = 0 . The initial value a0 obtained from this equation will be the critical value separating the solutions which increase without bound with t ? 0 from those which decrease without bound with t ? 0.

The answer I typed in is but I don't know how to come to that answer
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September 13th, 2013, 09:46 AM   #6
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Re: Diff Eq - Solve the initial value problem

This makes little sense to me. As we should expect . It is the behavior as that is affected by the choice of .
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September 13th, 2013, 11:02 AM   #7
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Re: Diff Eq - Solve the initial value problem

Quote:
Originally Posted by MarkFL
This makes little sense to me. As we should expect . It is the behavior as that is affected by the choice of .
Alright. You've been a big help. Thanks a lot
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