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

This is a first-order linear differential equation, which can be solved using an integrating factor.

First, we need to rearrange the equation in the standard form:

dy/dx + py = -q

Now, we can find the integrating factor, which is given by:

IF = e^(∫pdx)

Integrating p with respect to x, we get:

IF = e^(∫pdx) = e^(px)

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

e^(px)dy/dx + e^(px)py = -e^(px)q

The left-hand side can be written as the product rule of (e^(px)y):

(d/dx)(e^(px)y) = -e^(px)q

Integrating both sides with respect to x, we get:

e^(px)y = -∫e^(px)qdx + C

where C is the constant of integration.

Finally, solving for y, we get:

y = -e^(-px)∫e^(px)qdx + Ce^(-px)

This is the general solution to the differential equation.