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Math Help - Please Help! Ordinary/inital and second order differential equation.

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    Please Help! Ordinary/inital and second order differential equation.

    Hi
    The first question ordinary differential equation whats putting me off, is moving the x's and y's to one side and the cos(x) doesnt help or isnt this needed to intergrate?
    The second question not sure how to solve the quadratic equation r^2-r-12
    The last question =3e^5t?

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    Last edited by jjssj; April 23rd 2012 at 06:21 PM.
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    Re: Please Help! Ordinary/inital and second order differential equation.

    Quote Originally Posted by jjssj View Post
    The first question ordinary differential equation whats putting me off, is moving the x's and y's to one side and the cos(x) doesnt help or isnt this needed to intergrate?
    First thing to do is divide both sides by \cos x

    Quote Originally Posted by jjssj View Post

    The second question not sure how to solve the quadratic equation r^2-r-12
    Find factors of -12 to sum to -1.
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    Re: Please Help! Ordinary/inital and second order differential equation.

    is this right for the first question
    dy+y=(cos^3xsinx)dx/sinx can intergrate this now?
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    Re: Please Help! Ordinary/inital and second order differential equation.

    Quote Originally Posted by jjssj View Post
    is this right for the first question
    dy+y=(cos^3xsinx)dx/sinx can intergrate this now?
    No, it should be

    \displaystyle \begin{align*} \cos{(x)}\,\frac{dy}{dx} + \sin{(x)}\,y &=  2\cos^3{(x)}\sin{(x)} - 1 \\ \frac{dy}{dx} + \frac{\sin{(x)}}{\cos{(x)}}\,y &= \frac{2\cos^3{(x)}\sin{(x)} - 1}{\sin{(x)}} \end{align*}

    And this can be solved by multiplying both sides by the integrating factor \displaystyle \begin{align*} e^{\int{\frac{\sin{(x)}}{\cos{(x)}}}\,dx} = e^{-\ln{\left[\cos{(x)}\right]}} = e^{\ln{\left\{\left[\cos{(x)}\right]^{-1}\right\}}} = \frac{1}{\cos{(x)}} \end{align*}
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