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

This is a first-order linear differential equation of the form:

dy/dx + py + q = 0

To solve this equation, we need to use an integrating factor. The integrating factor is defined as e^(int p dx), where int p dx is the integral of p with respect to x.

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

e^(int p dx) dy/dx + e^(int p dx) py + e^(int p dx) q = 0

The left-hand side of the equation can be written as the derivative of the product of the integrating factor and y:

d/dx (e^(int p dx) y) + e^(int p dx) q = 0

Integrating both sides with respect to x, we get:

e^(int p dx) y = -int e^(int p dx) q dx + C

where C is the constant of integration.

Finally, solving for y, we get:

y = -1/p (e^(-int p dx) int e^(int p dx) q dx + Ce^(-int p dx))

This is the general solution to the differential equation dy/dx + py + q = 0.