How do you separate variables in the differential equation $\cos x \sin y dy = \cos y \sin x dx$?

Conditions

• The equation is a first-order differential equation.
• cos x and cos y are not zero at the points of interest, allowing division.

Reasoning, step by step

1. Start with the equation: $\cos x \sin y dy = \cos y \sin x dx$.
2. Identify the factors that need to be moved: cos x on the left and cos y on the right.
3. Divide both sides by cos x cos y.
4. Simplify the expression to get (sin y)/(cos y)$dy = (\sin x)/(\cos x)dx$.
5. Recognize that the variables are now fully separated.

Example

The instructor explicitly performs this division on the board to transform the trigonometric equation into a separable form.

Common misconceptions

• Dividing by only one of the cosine terms, failing to fully separate the variables.
• Forgetting that dividing by cos x cos y affects both sides of the equation simultaneously.

Watch the explanation

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• How do you integrate both sides of (sin y)/(cos y)$dy = (\sin x)/(\cos x)dx$?
• Why is this equation considered nonlinear?