Solving First-Order Linear Differential Equations: dy/dx + py + q = 0

This is a first-order linear differential equation of the form:

dy/dx + p(x)y = -q(x)

To solve this equation, we use an integrating factor, which is defined as:

I(x) = e^(∫p(x)dx)

Multiplying both sides of the equation by the integrating factor, we get:

e^(∫p(x)dx) dy/dx + e^(∫p(x)dx) p(x) y = -e^(∫p(x)dx) q(x)

The left-hand side can be rewritten using the product rule:

d/dx ( e^(∫p(x)dx) y ) = -e^(∫p(x)dx) q(x)

Integrating both sides with respect to x, we get:

e^(∫p(x)dx) y = ∫-e^(∫p(x)dx) q(x) dx + C

Dividing both sides by the integrating factor, we obtain the general solution:

y(x) = e^(-∫p(x)dx) ( ∫-e^(∫p(x)dx) q(x) dx + C )

where C is the constant of integration.