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Math Help - separable equations (DFQ)

  1. #1
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    separable equations (DFQ)

    Alright so I have this equation and I need to solve it using the separable equation method ...

    sin(x)dy/dx + ycos(x) = xsin(x)

    I tried to separate the equation and got this ...

    (1/y)dy = cos(x)/(sin(x)-xsin(x))dx

    but I cannot integrate the right side of the equation. I am not sure if I separated them incorrectly or if I am missing something on integrating it.

    Any help is appreciated!
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  2. #2
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    Quote Originally Posted by cheertcc101 View Post
    Alright so I have this equation and I need to solve it using the separable equation method ...

    sin(x)dy/dx + ycos(x) = xsin(x)

    I tried to separate the equation and got this ...

    (1/y)dy = cos(x)/(sin(x)-xsin(x))dx

    but I cannot integrate the right side of the equation. I am not sure if I separated them incorrectly or if I am missing something on integrating it.

    Any help is appreciated!
    If you divide through the original equation by sin(x) you get:

     \frac{dy}{dx} + y\cot(x) = x
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  3. #3
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    From there how do I get the y and the dy on the same side of the equation so I can integrate??
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  4. #4
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    Quote Originally Posted by cheertcc101 View Post
    Alright so I have this equation and I need to solve it using the separable equation method ...

    sin(x)dy/dx + ycos(x) = xsin(x)

    I tried to separate the equation and got this ...

    (1/y)dy = cos(x)/(sin(x)-xsin(x))dx

    but I cannot integrate the right side of the equation. I am not sure if I separated them incorrectly or if I am missing something on integrating it.

    Any help is appreciated!
    Hint: (y\sin x )' = x\sin x
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  5. #5
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    Alright so the derivative of ysinx = xsinx .. but Im not sure how this helps me at all and also I am not sure that the derivative of ysinx = xsinx so maybe I dont understand what you mean?
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  6. #6
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    My bad it was a linear equation and not separable SORRY but THANKS
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  7. #7
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    Quote Originally Posted by cheertcc101 View Post
    My bad it was a linear equation and not separable SORRY but THANKS
    Your equation is in the form:

     \frac{dy}{dx} + p(x)y = r(x)

    The solution to such an equation is given by:

     \phi(x)y(x) = \int r(x)\phi(x) dx

    Where  \phi(x) is the integrating factor. In this case  \phi(x) = e^{\int p(x)dx} = e^{\int \text{cot}(x)dx}  =e^{\ln|\sin(x)|} = \sin(x)

    Hence:

     \sin(x)y(x) = \int x\sin(x) dx

     \sin(x)y(x) =  [-x\cos(x)] - \int -cos(x) dx

     \sin(x)y(x) =  -x\cos(x) + \sin(x)+C

     y(x) =  -x\text{cot}(x) + 1+\frac{C}{\sin(x)}
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