First order differential equation

[magnifik]

find the solution for the following equation:

dy(t)/dt + 3y(t) = x(t-2), t > 0 and y(0) = 0

i did it by parameterization but am unsure if it is correct. i need help especially with the limits of integration.

y' + 3y = x(t-2)
y_h' = -3y_h
y_h = Ce^-3t
y(t) = v(t)e^-3t
(ve^-3t)' = v'e^-3t - 3ve^-3t
-3ve^-3t + v'e^-3t = -3ve^-3t + x(t-2)
v' = e^3t * x(t-2)
v = int[e^3t * x(t-2) dt] // int means integral
y(t) = e^-3t * int[e^3T * x(T-2) dT] // T is tau
i'm not sure if that's correct. right now i have the limits of integration set to 0 to t

thanks in advance

 

[hunt_mat]

One way is to take Laplace transforms. I am assuming here that x(t) is a predefined function

 

[magnifik]

we have not discussed Laplace transforms in class yet. x(t-2) stands alone. no further information is given.