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 it, we need to use an integrating factor, which is given by:

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 can be rewritten using the product rule as:

d(IF(x)y)/dx = 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 a constant of integration.

Dividing both sides by IF(x), we obtain the solution for y:

y = (1/IF(x))∫q(x)IF(x)dx + C/IF(x)

Substituting the value of IF(x), we get:

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

This is the general solution to the differential equation. The constant C can be determined using initial conditions.