Final answer:
To find g(t), we use the Wronskian of f and g, differentiate it, and solve a second-order linear homogeneous differential equation. The solution for g(t) is g(t) = C1 * e^t + C2 * e^(-t) + (1/24)t^3 * e^8t.
Step-by-step explanation:
To find g(t), we can use the fact that the Wronskian of f and g is given by W = f * g' - f' * g, where f' and g' are the derivatives of f and g, respectively.
In this case, f(t) = t, so f'(t) = 1. Now we can differentiate W with respect to t, using the product rule:
W' = f' * g' + f * g'' - f'' * g - f' * g' = 8t^2 * e^8t
Since f'(t) = 1, we can simplify this equation to:
g'' - g = 8t^2 * e^8t
This is a second-order linear homogeneous differential equation. The general solution is of the form g(t) = C1 * e^t + C2 * e^(-t). In this case, we need to find the particular solution. We can use the method of undetermined coefficients to guess a particular solution of the form g(t) = At^3 * e^8t.
Substituting this into the equation, we get:
192At^3 * e^8t = 8t^2 * e^8t
From this, we can solve for A and find that A = 1/24.
Therefore, the solution for g(t) is g(t) = C1 * e^t + C2 * e^(-t) + (1/24)t^3 * e^8t.