Put the differential equation in the initial form dy/dt + p(t) y = g(t). This is a standard form which will help you solve the problem.
Find the integrating factor, e^S p(t) dt, where S stands for the integral sign. Integrate the power on e and rewrite the integrating factor with the new function as its power.
Multiply all terms on both sides of the equation by the integrating factor. Note that the left-hand side becomes the product rule, ((e^S p(t) dt) * y(t))', and write it in this form.
Integrate both sides of the equation. Don't forget to add a constant of integration, +c, to the right-hand side of the equation. (You could add a different arbitrary constant to each side of the equation, then subtract the one on the left, but the result is the same, since c is arbitrary anyway.)
Solve the equation for y(t). Your answer will still contain the arbitrary constant, c, which can only be solved for a specific value if you were given some initial conditions to plug into the equation.