Final answer:
To find r(t) for the given conditions, we can integrate r'(t) = 4e^(2t)i + 3e^tj. By performing the integration and substituting the initial condition, r(0) = 2i, we find that r(t) = (2e^(2t) - 2)i + 3e^tj.
Step-by-step explanation:
To find r(t) for the given conditions, we need to integrate r'(t). Given that r'(t) = 4e^(2t)i + 3e^tj and r(0) = 2i, we can proceed as follows:
Integrate each component separately:
∫ 4e^(2t) dt = 2e^(2t) + C1
∫ 3e^t dt = 3e^t + C2
Since r(0) = 2i, we can substitute t = 0 in the integrated components:
2e^0 + C1 = 2i
3e^0 + C2 = 0j
From the second equation, we find that C2 = 0. Substituting this value in the first equation, we find that C1 = 2i - 2 = -2 + 2i.
Finally, we substitute the values of C1 and C2 back into the integrated components:
r(t) = (2e^(2t) - 2)i + 3e^tj