
Calculus Calculus Math Forum 
 LinkBack  Thread Tools  Display Modes 
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  
Member Joined: Jun 2012 From: San Antonio, TX Posts: 84 Thanks: 3 Math Focus: Differential Equations, Mathematical Modeling, and Dynamical Systems  Re: Diff Eq  Solve the initial value problem Quote:
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  
Member Joined: Jun 2012 From: San Antonio, TX Posts: 84 Thanks: 3 Math Focus: Differential Equations, Mathematical Modeling, and Dynamical Systems  Re: Diff Eq  Solve the initial value problem Quote:
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  
Member Joined: Jun 2012 From: San Antonio, TX Posts: 84 Thanks: 3 Math Focus: Differential Equations, Mathematical Modeling, and Dynamical Systems  Re: Diff Eq  Solve the initial value problem Quote:
 

Tags 
diff, initial, problem, solve 
Thread Tools  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
Solve for initial Value  rsoy  Calculus  2  November 2nd, 2012 04:37 AM 
solve the given diff eq. by the method of undetrmined coeff.  reggea07  Applied Math  1  May 19th, 2011 02:56 PM 
solve the initial value problem  rsoy  Calculus  18  December 21st, 2010 01:46 PM 
SOLVE THE DIFF.EQUATION  zgonda  Calculus  1  November 27th, 2010 04:23 PM 
Help me to solve a problem in Diff Geometry  kien  Applied Math  0  April 12th, 2009 09:44 AM 