1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Initial value differential

  1. Aug 28, 2013 #1
    Problem: Find the solution to the initial Value problem (differential equations)?

    y(0) = 3

    Attempt: y' -y = e^(-2x) ----- (1)

    Linear equation of first order of form y' + y p(x) = q(x)
    p(x)= -1
    q(x) = e^(-2x)

    Find the integrating factor e^∫p(x) dx = e^∫ - dx = e^-x

    Multiply equation (1) by the integrating factor e^-x
    y' e^-x - y e^-x = e^(-3x) ----- (2)

    The left-hand side of (2) is the derivative of y times the integrating factor or (y e^-x)'
    (y e^-x)' = e^(-3x) ----- (3)

    Integrate both sides of (3)
    y e^-x = ∫ e^(-3x)
    y e^-x = (-1/3) e^(-3x) + C
    multiply everything by e^x

    y = (-1/3) e^(-2x) + C
    3 = (-1/3) e^0 + C
    C = 3+1/3 = 10/3

    y = (-1/3) e^(-2x) + (10/3) e^x

    How does this look?
  2. jcsd
  3. Aug 28, 2013 #2


    User Avatar
    Homework Helper

    The first thing you should do is to substitute your solution back into the d.e. Does it solve the equation?

    Is this ##y' = -y + e^{-2x}## or is this ##y' - y = e^{-2x}## your equation?

    Because in the problem statement, you gave the former, but you solved the latter. The solution for the latter is correct, but if you made a simple algebraic error in bringing the ##y## over, then the solution is obviously wrong.
    Last edited: Aug 28, 2013
  4. Aug 28, 2013 #3


    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    Isn't C included in the "everything"?
  5. Aug 28, 2013 #4


    User Avatar
    Homework Helper

    Yes, it should be. The reason it didn't affect the answer is because ##e^0 = 1##.
  6. Aug 29, 2013 #5


    User Avatar
    Science Advisor
    Gold Member
    2017 Award

    Why not using the standard way of solving linear equations, i.e., first solving the homogeneous equation
    and then using the ansatz of the variation of the constant?
  7. Aug 29, 2013 #6


    User Avatar
    Homework Helper

    The integrating factor is the usual elementary method that's taught for questions like this, I think.

    I was about to suggest using a Laplace transform, which gives a quick algebraic solution in an initial value problem like this, but thought better of it, simply because it's unlikely the thread starter has covered it yet.
    Last edited: Aug 29, 2013
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted