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Thread: dy/dx - y = sin x ( form differential equation)

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    dy/dx - y = sin x ( form differential equation)

    i found that the integrating factor is (e^-x ) , so i gt y (e^-x ) = integral of sin x (e^-x ) , how to integrate sin x (e^-x ) ?
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    Re: dy/dx - y = sin x ( form differential equation)

    Quote Originally Posted by xl5899 View Post
    i found that the integrating factor is (e^-x ) , so i gt y (e^-x ) = integral of sin x (e^-x ) , how to integrate sin x (e^-x ) ?
    Here's a dandy little trick that you'll find useful every now-and-again. Do this by parts:
    $\displaystyle I = \int e^{-x}~sin(x)~dx = -e^{-x}~sin(x) + \int e^{-x}~cos(x)~dx$

    Do parts again on the second integral:
    $\displaystyle I = \int e^{-x}~sin(x)~dx = -e^{-x}~sin(x) + \left [ -e^{-x}~cos(x) - \int e^{-x}~sin(x)~dx \right ] $

    Notice that the last integral is just the same as the first integral. We can replace it with "I".
    $\displaystyle I = \int e^{-x}~sin(x)~dx = -e^{-x}~sin(x) - e^{-x}~cos(x) - I $

    Now finish by solving for I.

    -Dan
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