Final answer:
The correct answer is option is d. The differential equation u" + 36u = 8e3t is a second-order linear non-homogeneous differential equation. The general solution is a combination of the solutions derived from the corresponding homogeneous equation and a particular solution. The correct general solution is 8ut = C1cos6t + C2sin6t + 45/8e3t.
Step-by-step explanation:
The differential equation in question is u" + 36u = 8e3t. To solve this, we need to find the homogeneous solution and particular solution. The characteristic equation for the homogeneous part is r2 + 36 = 0, which gives us the roots r = ±6i. The homogeneous solution will therefore be of the form uh(t) = C1cos(6t) + C2sin(6t).
For the non-homogeneous part, we assume a particular solution of the form up(t) = A × e3t. Substituting up into the differential equation, we get A and thus find the particular solution. Combining both gives us the general solution of the equation, which includes terms from both the homogeneous and particular solution.
The correct option that represents the general solution is therefore d. 8ut = C1cos6t + C2sin6t + 45/8e3³¹t, where C1 and C2 are constants determined by initial conditions.