Answer:
To solve the differential equation u" - 4u' - 12u = 3e^5t, we can use the method of undetermined coefficients.
First, let's find the general solution to the homogeneous equation u" - 4u' - 12u = 0. To do this, we assume a solution of the form u(t) = e^(rt), where r is a constant.
Plugging this solution into the homogeneous equation, we get the characteristic equation:
r^2 - 4r - 12 = 0.
Solving this quadratic equation, we find the roots r1 = 6 and r2 = -2.
Therefore, the general solution to the homogeneous equation is:
u_h(t) = C1e^(6t) + C2e^(-2t), where C1 and C2 are arbitrary constants.
To find the particular solution to the nonhomogeneous equation, we assume a solution of the form u_p(t) = Ae^(5t), where A is a constant.
Plugging this particular solution into the nonhomogeneous equation, we get:
(25A - 20A - 12A)e^(5t) = 3e^(5t).
Simplifying, we find:
-7A = 3.
Solving for A, we get A = -3/7.
Thus, the particular solution is:
u_p(t) = (-3/7)e^(5t).
The general solution to the nonhomogeneous equation is the sum of the homogeneous and particular solutions:
u(t) = u_h(t) + u_p(t) = C1e^(6t) + C2e^(-2t) - (3/7)e^(5t).
And that is the solution to the given ordinary differential equation.