Final answer:
After separating and integrating the given differential equation y' = 2y/3x and applying the initial condition (8, 2), none of the provided options are correct. The specific solution is y = (2/e^(16/3))e^(2/3x), which does not match any of the answer choices.
Step-by-step explanation:
The given differential equation is y' = 2y/3x. We are looking for a solution that passes through the point (8, 2). This equation is separable, so we can write it as dy/y = (2/3)dx/x and then integrate both sides. The general solution of this equation is y = Ce^(2/3x), where C is a constant that we can find using the initial condition given by the point (8,2).
To find the constant C, we plug in x = 8 and y = 2 into the general solution:
2 = Ce^(2/3(8))
C = 2/e^(16/3)
The specific solution that goes through the point (8, 2) is y = (2/e^(16/3))e^(2/3x).
Comparing with the provided options, none of them match exactly with the standard form of the specific solution we found. The closest option is A) y = 2e^(2/3x), which would be correct if x = 0 and y = 2; however, given the initial condition (8, 2), this option does not satisfy it when plugged into the equation. Therefore, all options are incorrect under the given initial condition.