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 September 12th, 2013, 06:54 PM #1 Member   Joined: Jun 2012 From: San Antonio, TX Posts: 84 Thanks: 3 Math Focus: Differential Equations, Mathematical Modeling, and Dynamical Systems 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 September 12th, 2013, 07:44 PM #2 Senior Member   Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,211 Thanks: 521 Math Focus: Calculus/ODEs Re: Diff Eq - Solve the initial value problem Using essentially the same method, I get: 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. September 12th, 2013, 10:25 PM #4 Senior Member   Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,211 Thanks: 521 Math Focus: Calculus/ODEs 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: 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 September 13th, 2013, 09:46 AM #6 Senior Member   Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,211 Thanks: 521 Math Focus: Calculus/ODEs 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 . September 13th, 2013, 11:02 AM   #7
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Posts: 84
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Math Focus: Differential Equations, Mathematical Modeling, and Dynamical Systems
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 Tags diff, initial, problem, solve ,

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