Find the particular solution of the differential equation dy

Find the particular solution of the differential equation dy/dx + y cos(x) = 2 cos (x) satisfying the initial condition y(0) = 4. Your answer should be a function of x.

Solution

This equation is separable.

Subtract the second terms from the left-hand side and factor out cos (x):

dy/dx = (2-y) cos(x), so

(1/(2-y)) dy = cos(x) dx.

Integrate:

ln(2-y) = -sin(x) + c1

Exponentiate (use e^(both sides)):

2 - y = e^(-sin(x) + c1)

2 - y = e^(-sin(x)) e^(c1)

2 - y = c2*e^(-sin(x)) (where c2=e^(c1))

y - 2 = -c2*e^(-sin(x))

y = 2 - c2*e^(-sin(x)) (general solution)

OK. Since y(0) = 4, substitute in x = 0, y = 4:

4 = 2 - c2*e^(-sin(0))

4 = 2 - (c2 * 1)

so c2 = -2, and the particular solution is

y = 2 + 2*e^(-sin(x)).