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

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

dy/dx + py = -q

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

μ(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) py = -e^(∫p(x)dx) q

The left-hand side of this equation can be simplified using the product rule of differentiation:

d/dx (e^(∫p(x)dx) y) = -e^(∫p(x)dx) q

Integrating both sides with respect to x, we get:

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

where C is the constant of integration.

Finally, solving for y, we get:

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

This is the general solution to the first-order linear differential equation dy/dx + py = -q.