To solve the differential equation csc(y) dx + sec²(x) dy = 0 by separation of variables, we can rearrange it into a form where each variable can be isolated.
First, we rewrite the equation:
csc(y) dx = -sec²(x) dy
Now, we can separate the variables. We will move all the terms involving y to one side and the terms involving x to the other side:
csc(y) dy = -sec²(x) dx
Next, we can integrate both sides:
∫ csc(y) dy = -∫ sec²(x) dx
The integral of csc(y) is -ln|csc(y) + cot(y)| + C, and the integral of sec²(x) is tan(x). So we get:
-ln|csc(y) + cot(y)| = -tan(x) + C
Multiplying through by -1, we have:
ln|csc(y) + cot(y)| = tan(x) – C
To express the solution, we can exponentiate both sides, which gives us:
|csc(y) + cot(y)| = e^{tan(x) – C}
We can denote e^{-C} as a new constant, say K, which leads us to:
csc(y) + cot(y) = K e^{tan(x)}
This is the general solution of the given differential equation. By applying initial conditions, if any are provided, we can find the particular solution. However, without such conditions, this expression provides a complete solution.