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:

u(x) = e^(integral p(x) dx)

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

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

Using the product rule on the left-hand side, we can write:

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

Integrating both sides with respect to x, we get:

u(x)y = -integral q(x)u(x) dx + C

Dividing both sides by u(x), we get the solution for y:

y = (-1/u(x))integral q(x)u(x) dx + C/u(x)

Substituting the expression for u(x) in terms of p(x), we get:

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

This is the general solution for the given differential equation.