A level C4 INTEGRATION

We just finished C4 integration and I've been doing some questions from the textbook on separating variables and I've been struggling on this particular questions:
It says to find the general formula of the following differential equations:
1. sin y cosx dy/dx = xcos y / cosx
2. cos y sin2x dy/dx = cotx cosec y

Omg we haven't even finished vectors, how have u already finished integration? Is it really hard? What applied module r u doing )

What have you tried so far?

For 1.

sin y cos x dy/dx = x cos y /cos x
sin y cos x dy/dx = x/cos x * cos y
integral of sin y / cos y dy = integral of x/cos^2 x dx
integral of tan y dy = integral of x/ cos^2 x dx

That's all I could do for Q.1

integral of cos y/ (1/sin y) dy= integral of (cos x/sinx)/(1/2sinxcosx) dx
integral of cos y siny dy = integral of 1/2sin^2 x dx

Looks good to me.

The integral of tan(y) is given in the formula booklet, I believe (to derive it, write tan(y) in terms of sin(y) and cos(y) and play around but there's no need for this question).

You can write the right hand side as x sec^2x and use by parts to integrate that.



Firstly, sin2x = 2sinxcosx, not 1/2.

When you fix that, you may wish to have a look at the RHS again for the bit quoted above. Did you mean $\dfrac{1}{2}sin^2x$ ? If so, how did you get from the first line to the second as that's not correct?

Or for the RHS, did you mean $\dfrac{1}{2sin^2x}$ ? If so, you can write that as $\dfrac{1}{2}\csc^2x$ which you can integrate.

To integrate the LHS though, think of the double angle formulae and how else you can re-write it.