Question

Dy/dx = (sin x)/y , y(0) = 2

Answer 1

Start with

$\dfrac{dy}{dx}=\dfrac{\sin(x)}{y}$

Separate the variables:

$y\;dy = \sin(x)\;dx$

Integrate both parts:

$\displaystyle \int y\;dy = \int\sin(x)\;dx$

Which implies

$\dfrac{y^2}{2} = -\cos(x)+c$

Solving for y:

$y = \sqrt{-2\cos(x)+c}$

Fix the additive constant imposing the condition:

$y(0) = \sqrt{-2+c}=2 \iff -2+c=4 \iff c=6$

So, the solution is

$y(x) = \sqrt{-2\cos(x)+6}$