How to Solve the Differential Equation cosec y dx + sec 2x dy = 0 Using Separation of Variables

To solve the differential equation cosec y dx + sec 2x dy = 0 using separation of variables, we first rearrange the equation.

Starting with:

cosec y dx + sec 2x dy = 0

We can isolate the terms involving y and x. Rearranging gives:

cosec y dx = -sec 2x dy

Now, we separate variables:

cosec y dy = -sec 2x dx

Next, we integrate both sides. The left side can be integrated by recognizing that:

∫cosec y dy = -ln |cosec y + cot y| + C

And for the right side, we can use the identity:

sec 2x = 1/cos 2x

which allows us to write:

∫sec 2x dx = (1/2) ln |sec 2x + tan 2x| + C

Putting it together gives us:

-ln |cosec y + cot y| = - (1/2) ln |sec 2x + tan 2x| + C

We can then exponentiate both sides to remove the logarithms:

|cosec y + cot y| = K |sec 2x + tan 2x|^{-1/2}

where K = e^{-C}. Finally, we have our solution in a separated form that relates y and x.

This shows how the method of separation of variables applies to this trigonometric differential equation effectively.