Final answer:
To solve the initial value problem u''(x) = 48e^(4x) - 4e^(-2x), assume the solution has the form u(x) = Ae^(4x) + Be^(-2x). Differentiate twice and substitute back into the equation. Compare coefficients to find A = 3 and B = -1. Hence, the solution is u(x) = 3e^(4x) - e^(-2x).
Step-by-step explanation:
To solve the initial value problem u''(x) = 48e^(4x) - 4e^(-2x), we can use the method of undetermined coefficients. Let's assume that the solution has the form u(x) = Ae^(4x) + Be^(-2x). Differentiating u(x) twice, we have u''(x) = 16Ae^(4x) + 4Be^(-2x).
Substituting these expressions back into the original differential equation, we get 16Ae^(4x) + 4Be^(-2x) = 48e^(4x) - 4e^(-2x). By comparing the coefficients of different powers of e, we can deduce that A = 3 and B = -1.
Therefore, the solution to the initial value problem is u(x) = 3e^(4x) - e^(-2x).