Step-by-step explanation:
Certainly! Let's find the particular solution to the given differential equation using the method of undetermined coefficients.
The differential equation is: [y’’ + 7y’ + 12y = -680 \sin(t)]
First, let’s find the complementary solution (homogeneous part) of the differential equation. The characteristic equation is: [r^2 + 7r + 12 = (r - 6)(r + 2) = 0] Solving for the roots: [r_1 = -2, \quad r_2 = 6]
The complementary solution is: [y_c(t) = c_1 e^{-2t} + c_2 e^{6t}]
Now, let’s guess the particular solution in the form: [Y_P(t) = A e^{5t}]
Taking derivatives: [Y_P’(t) = 5A e^{5t}] [Y_P’'(t) = 25A e^{5t}]
Substitute into the differential equation: [25A e^{5t} + 7(5A e^{5t}) + 12(A e^{5t}) = -680 \sin(t)]
Solving for the coefficient (A): [25A - 35A + 12A = -680] [-7A = -680] [A = \frac{680}{7} = 97.14]
Therefore, the particular solution is: [Y_P(t) = 97.14 e^{5t}]
The complete solution is the sum of the complementary and particular solutions: [y(t) = y_c(t) + Y_P(t) = c_1 e^{-2t} + c_2 e^{6t} + 97.14 e^{5t}]
