I am reading an article, which i have to do a similar work. But my knowledge in maths is not advanced. There are two time series equation first: \begin{align} x(t) &= \sin(wt) \qquad \tag{ equation 1} \\ y(t) &= \sin(wt + \phi) \tag{ equation 2} \\ y(t) &= \sin(wt)\cos(\phi) + \cos(wt)\sin(\phi) \tag{ equation 3} \end{align}
using equation $1$, $t = (1/w)\arcsin(x)$, replacing this in equation 3, gives equation 4: $$ y = x\cos(\phi) + \cos[\arcsin(x)]\sin(\phi) \qquad \tag{ equation 4}$$
Then, the author just produce the next equation by saying, he derived it from equation 3 by doing an ordinary differential equation in y by eliminating time, thus he got the following:
$$dy(t)/dt - y(t)[w\cot(\phi)] = -w\csc(\phi)x(t) \tag{ equation 5}$$
My question is I am unable to obtain equation $5$ and I don't understand how he did it. Did he used equation $4$ instead of $3$ to get it as we have the $x$ variable in equation.
Can someone please write the full derivation of equation $5$ from the above equations for me.
note: csc is cosecant and cot is cotangent
Thank you.