Solve the differential equation by separation of variables: $csc\left(y\right) dx+{sec}^2\left(x\right) dy=0$. Which of the following represents the general solution?

• A. $sin\left(y\right)-tan\left(x\right)=C$
• B. $cos\left(y\right)+tan\left(x\right)=C$
• C. $cos\left(y\right)-tan\left(x\right)=C$
• D. $sin\left(y\right)+tan\left(x\right)=C$

Answer: (D)

Verified step by step guidance

1. Step 1: Rewrite the given differential equation csc(y) dx + sec^2(x) dy = 0 in a form suitable for separation of variables. Move terms involving dx and dy to separate sides: csc(y) dx = -sec^2(x) dy.
2. Step 2: Divide both sides by csc(y) sec^2(x) to isolate dx and dy terms: dx / sec^2(x) = -dy / csc(y).
3. Step 3: Integrate both sides separately. For the left-hand side, integrate ∫(cos^2(x)) dx, and for the right-hand side, integrate ∫(-sin(y)) dy.
4. Step 4: After performing the integrations, you will obtain expressions involving trigonometric functions. Combine the results into a single equation representing the relationship between x and y.
5. Step 5: Introduce the constant of integration (C) to represent the general solution. Rearrange the equation to match the form sin(y) + tan(x) = C, which is the correct answer.