1. ## 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. Originally Posted by limegreen1545
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:

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

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

Now apply the initial condition to find C.

Can you continue from here?