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Differential equation dy/dx = 2y-4x

  1. Mar 30, 2009 #1
    1. The problem statement, all variables and given/known data

    Consider the differential equation dy/dx = 2y-4x

    a. Find the value of b for which y=2x+b is a solution to the given differential equation

    b.Let g be the function that satisfies the given differential equation with the initial condition g(0)=0. Does the graph of g have a local extremum at the point (0,0) ? If so, is the point a local maximum or minimum

    2. Relevant equations

    integration, derivative

    3. The attempt at a solution

    a. So I choose point (1,0) and plug in y=2x+b and solve for b=1. Then I have y=2x+1. AM I right ? It seems too easy to be right. Is it harder than I thought ?

    b. So do I have to integrate ? I found that (0,0) is a critical point. How do I move on ?
     
  2. jcsd
  3. Mar 30, 2009 #2

    rock.freak667

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    Homework Helper

    For part a) what you need to do is find dy/dx from your given solution and then substitute y in the differential equation and find b. Can you do that?

    It would be beneficial to solve the differential equation.
     
  4. Mar 30, 2009 #3
    a. Got cha.

    b. is g(x)= 2x+1 ? Am I right ? It sounds wrong
     
  5. Mar 30, 2009 #4

    rock.freak667

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    Homework Helper

    Do you know how to solve 1st order DEs of the form
    [tex]\frac{dy}{dx}+P(x) y=Q(x)[/tex]

    via an integrating factor?
     
  6. Mar 30, 2009 #5
    Yeah. so is the integrating factor only e^2 ?
    I got the final equation as y=-2x^2. Am I right ?
     
  7. Mar 30, 2009 #6
    Also a question about series:

    Given a McLauren series: (2x)^n+1 / (n+1)

    (a). Find interval of convergence.

    So I used ratio test and found that -1/2 <x<1/2. I am testing the end point. At x=1/2, the series will be 1/(n+1) and at x=-1/2, series is (-1)^n+1 / (n+1). How do I prove whether or not they are divergent or convergent. Does 1/ (n+1) converge to 0 ?
     
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