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 can use an integrating factor, which is defined as:

IF(x) = e^(∫p(x)dx)

Multiplying both sides of the equation by IF(x), we get:

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

The left-hand side is the product rule for differentiation of (IF(x)y), so we can rewrite the equation as:

d/dx(IF(x)y) = -q(x)IF(x)

Integrating both sides with respect to x, we get:

IF(x)y = ∫-q(x)IF(x)dx + C

where C is the constant of integration.

Finally, solving for y, we get:

y = (1/IF(x)) * (∫-q(x)IF(x)dx + C)

Substituting the expression for IF(x), we get:

y = (1/e^(∫p(x)dx)) * (∫-q(x)e^(∫p(x)dx)dx + C)

This is the general solution to the given differential equation.