Solving First-Order Linear Homogeneous Differential Equations (dy/dx + py + q = 0)

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

Step 1: Rewrite the equation in standard form.

dy/dx + py = -q

Step 2: Find the integrating factor (IF) by multiplying both sides of the equation by e^(integral of p dx).

IF = e^(integral of p dx)

Step 3: Multiply both sides of the equation by the integrating factor.

IF * dy/dx + p * IF * y = -q * IF

Step 4: Recognize the left-hand side as the product rule of (IF * y).

d/dx (IF * y) = -q * IF

Step 5: Integrate both sides with respect to x.

IF * y = -integral of q * IF dx + C

Step 6: Solve for y.

y = (-1/IF) * integral of q * IF dx + C/IF

Therefore, the general solution to the differential equation is:

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

where C is the constant of integration.