Find the particular solution of dy/dx = (y cos x)/(1 + sin x), given that y = 2 when x = 0.

1. 1 · Write what is given and what is to be found. Given: dy/dx = (y cos x)/(1 + sin x) and y(0) = 2. To find: the particular solution.
2. 2 · Write the method. The right side is a function of y times a function of x, so separate the variables and integrate.
3. 3 · Separate the variables. dy/y = cos x/(1 + sin x) dx
4. 4 · Integrate both sides.
   • ∫ dy/y = log|y|
   • Put t = 1 + sin x, dt = cos x dx: ∫ cos x/(1 + sin x) dx = log|1 + sin x|
   • log|y| = log|1 + sin x| + log|C|, so y = C(1 + sin x)
5. 5 · Use the condition. x = 0, y = 2: 2 = C(1 + 0), so C = 2.
6. 6 · Check and box the answer. y = 2(1 + sin x). Check: dy/dx = 2 cos x, and (y cos x)/(1 + sin x) = 2 cos x. Both sides match.

Scoring tip: Write the constant as log|C| when both integrals are logs. Then you can remove the logs in one clean line.