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Math Help - differential equations

  1. #1
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    differential equations

    can anyone please explain step by step how to solve this problem? I'm supposed to solve for where the question mark is and i do not understand at all what to do. So if anyone can give a clear detailed step by step explanation that would be awesome! thank you!

    Find the solution of the differential equation that satisfies the given initial condition.

    ?
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  2. #2
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by limegreen1545 View Post
    can anyone please explain step by step how to solve this problem? I'm supposed to solve for where the question mark is and i do not understand at all what to do. So if anyone can give a clear detailed step by step explanation that would be awesome! thank you!

    Find the solution of the differential equation that satisfies the given initial condition.

    ?
    Apply separation of variables:

    \frac{\,dy}{\,dx}=\frac{y\cos x}{8+y^2}\implies \left(\frac{8}{y}+y\right)\,dy=\cos x\,dx

    Integrate both sides to get 8\ln\!\left|y\right|+\tfrac{1}{2}y^2=-\sin x+C

    Now apply the initial condition to find C.

    Can you continue from here?
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