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

This is a first-order linear differential equation, where p and q are constants.

The general solution to this equation is given by:

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

where C is the constant of integration.

To solve for y(x), we first need to find ∫p dx and ∫q e^(∫p dx) dx.

Once we have these integrals, we can plug them into the general solution and solve for y(x).

Let's go through the steps:

1. Find ∫p dx:

This is the integral of the coefficient of y in the equation. In this case, p is the coefficient of y, so we need to integrate p with respect to x.

∫p dx = px + K

where K is the constant of integration.

2. Find ∫q e^(∫p dx) dx:

This is the integral of q multiplied by e to the power of the integral of p with respect to x. We can use substitution to solve this integral.

Let u = ∫p dx, then du/dx = p, and dx = du/p.

Substituting into the integral, we get:

∫q e^(∫p dx) dx = ∫q e^u (du/p)

= (1/p) ∫q e^u du

= (1/p) * q * e^u + C

= (1/p) * q * e^(∫p dx) + C

where C is the constant of integration.

3. Plug the integrals into the general solution:

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

= e^(-px-K) * ((1/p) * q * e^(px+K) + C)

= e^(-px) * (1/p) * (q + Ce^K)

where C and K are constants.

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