How can we find the general solution of the differential equation given by cos(x) dy/dx + sin(x) y = 1?

To solve the differential equation cos(x) dy/dx + sin(x) y = 1, we first rewrite it in a standard form. We start by rearranging the terms.

The given equation can be rewritten as:

dy/dx + (sin(x)/cos(x)) y = 1/cos(x)

This is a first-order linear ordinary differential equation (ODE) of the form:

dy/dx + P(x) y = Q(x)

where:

Next, we need to find an integrating factor, μ(x), which is given by:

μ(x) = e^{∫ P(x) dx} = e^{∫ tan(x) dx}

The integral of tan(x) is -ln|cos(x)|. Therefore, the integrating factor is:

μ(x) = e^-ln|cos(x)| = |cos(x)|^-1 = sec(x)

Multiplying the entire ODE by the integrating factor:

sec(x) dy/dx + sec(x) (tan(x) y) = 1

This simplifies to:

d/dx [sec(x) y] = 1

Now, we integrate both sides with respect to x:

∫ d/dx[sec(x) y] dx = ∫ 1 dx

Which gives us:

sec(x) y = x + C

Where C is the constant of integration. Finally, we solve for y:

y = (x + C) cos(x)

Thus, the general solution to the differential equation cos(x) dy/dx + sin(x) y = 1 is:

y = (x + C) cos(x)

where C is any constant.