Solve the given differential equation by separation of variables. `e^y sin 2xdx + cosx(e^2y -y) dy=0`

Solve the given differential equation by separation of variables. 

`e^y sin 2xdx + cosx(e^2y -y) dy=0`

Solution:

To solve the differential equation `e^y sin 2x dx + cos x (e^2y - y) dy = 0` by separation of variables, we first need to rewrite it in the form `f(x)dx = g(y)dy`. To do this, we can divide both sides by `e^y sin 2x`:

`e^y dy = (e^2y - y) cos x dx /sin 2x`

Now we can separate the variables by moving all the y terms to one side and all the x terms to the other side. This gives us:

`e^y dy = (e^2y - y) cos x dx /(2 sin x cos x)`

`e^y dy = (e^2y - y) dx/(2 cos x)`

Integrating both sides with respect to their respective variables gives:

`∫ e^y dy = ∫ (e^2y - y) dx/(2 cos x) +C`

Solving for y gives:

`e^y = C*2cosx + 1/e^x`

`e^y = Ce^x + 1/e^x`

`y = ln(Ce^x + 1/e^x)`

So, the general solution is `y = ln(Ce^x + 1/e^x)` where C is an arbitrary constant.