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

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

First, we need to rearrange the equation to have y on one side and dy/dx on the other side: dy/dx = -py - q

Next, we need to find an integrating factor, which is a function of x that makes the left-hand side of the equation the derivative of the product of the integrating factor and y:

Integrating factor = e^(integral of p(x) dx)

Multiplying both sides of the equation by the integrating factor, we get: e^(integral of p(x) dx) dy/dx + e^(integral of p(x) dx)py = -e^(integral of p(x) dx)q

Now, we can apply the product rule to the left-hand side: d/dx(e^(integral of p(x) dx) y) = -e^(integral of p(x) dx)q

Integrating both sides with respect to x, we get: e^(integral of p(x) dx) y = -integral of e^(integral of p(x) dx)q dx + C

where C is the constant of integration.

Finally, we can solve for y by dividing both sides by the integrating factor: y = (-integral of e^(integral of p(x) dx)q dx + C) / e^(integral of p(x) dx)

This is the general solution to the differential equation.