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Math Help - [SOLVED] find integrating factor and solve the equation 3

  1. #1
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    [SOLVED] find integrating factor and solve the equation 3

    y' + y = e^x ; y(0) = 1

    1st, i calculate the integrating factor...
    u(x) = e^x

    times the integrating factor with DE...

    y'e^x + ye^x = e^2x

    dy/dx e^x + ye^x = e^2x

    d/dx ye^x = e^2x

    ye^x = ∫ e^2x dx
    .........= 1/2 e^2x + C

    y = 1/2 e^x + C

    the problem here, i didn't get the answer given which is..
    y = 1/2 (e^x + e^-x)
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  2. #2
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by nameck View Post
    y' + y = e^x ; y(0) = 1

    1st, i calculate the integrating factor...
    u(x) = e^x

    times the integrating factor with DE...

    y'e^x + ye^x = e^2x

    dy/dx e^x + ye^x = e^2x

    d/dx ye^x = e^2x

    ye^x = ∫ e^2x dx
    .........= 1/2 e^2x + C

    y = 1/2 e^x + C

    the problem here, i didn't get the answer given which is..
    y = 1/2 (e^x + e^-x)
    Everything is fine up to here:

    e^xy=\tfrac{1}{2}e^{2x}+C.

    When you multiply both sides by e^{-x}, you get y=\tfrac{1}{2}e^x+Ce^{-x}.

    Applying the initial condition gives us C=\tfrac{1}{2}.

    As a result, the solution is y=\tfrac{1}{2}\left(e^x+e^{-x}\right).

    Does this make sense?
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  3. #3
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    Quote Originally Posted by Chris L T521 View Post
    Applying the initial condition gives us C=\tfrac{1}{2}.
    yup2!! thanks chris..
    however.. i still dont know at which step i should use the initial value given to solve C?
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  4. #4
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by nameck View Post
    yup2!! thanks chris..
    however.. i still dont know at which step i should use the initial value given to solve C?
    when you get y=\tfrac{1}{2}e^x+Ce^{-x}, you can apply the initial condition.

    Since y(0)=1, we have 1=\tfrac{1}{2}e^0+Ce^{-0}=\tfrac{1}{2}+C\implies C=\tfrac{1}{2}.

    In general, though, initial conditions can be applied once all the derivative terms in the differential equation disappear in your solution.
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  5. #5
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    got it!! Thanks Chris L T521
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  6. #6
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by nameck View Post
    got it!! Thanks Chris L T521
    Also, feel free to reference my DE tutorial. I talk about differential equations of this form. The link can be found in my signature.
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