Solving Equation: dy/dt + 3y = x(t-2) | Time Invariance

[magnifik]

find the solution for the following equation & determine if it is time invariant:

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)
yh' = -3yh
yh = 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
i know how to check for time-invariance/variance but cannot go on with this part unless i know the formula is right

thanks in advance

 

[Valkarie]

for your help.Yes, the solution you have derived is correct. The limits of integration should be 0 to t, since you are solving for y(t). The equation is time-invariant, since it does not depend on the value of t.