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Please check my work on initial value problem. thank you!

  1. Apr 18, 2010 #1
    1. The problem statement, all variables and given/known data
    Solve the following initial value problem. Sketch the solution and describe its behavior
    as t increases.
    y'' + 4y' + 3y = 0

    y(0) = 2
    y'(0) = -1

    1st i solved the characteristic:

    r^2 + 4r + 3 = 0

    r = -1
    r = -3

    then the general solution is;

    y = c_1e^-x + c_2e^-3x
    also..
    y' = -c_1e^-x - 3c_2e^-3x

    so after i plug in the initial values i get 2 equations and 2 unknowns..

    c_1 + c_2 = 2
    -c_1 - 3c_2 = -1

    c_1 = 5/2
    c_2 = -1/2

    so the solution with initial conditions (is this called "particular solution?) is:

    y = (5/2)e^-x - (1/2)e^-3x

    is this much correct before i describe the equations behavior? thanks alot

    2. Relevant equations



    3. The attempt at a solution
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Apr 18, 2010 #2

    Mark44

    Staff: Mentor

    You've done all the hard work. Checking the solution to an initial value problem is easy in comparison. For your solution, y = (5/2)e-x - (1/2)e-3x, is it true that y(0) = 2 and y'(0) = -1? If so, then your solution satisifies the initial conditions, mean that the solution function goes through (0, 2) and its slope there is -1.

    For your solution function, is it true that y'' + 4y' + 3y = 0? If so, then your solution satisifies the differential equation, and you're home free.
     
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