Question

Determine the solutions, if any, to the given boundary value problem.

y′′ +6y′ +18y=4e³ˣ cos(3x); y(0)=0, y(π)=0

Answer 1

To solve the given boundary value problem, we can use the method of undetermined coefficients.

Step-by-step explanation:

To solve the given boundary value problem, we can use the method of undetermined coefficients. First, we find the complementary solution, which is the solution to the homogeneous equation y'' + 6y' + 18y = 0. The characteristic equation is r^2 + 6r + 18 = 0, which has complex roots -3 + 3i and -3 - 3i.

The complementary solution is of the form y_c(x) = c1e^(-3x)cos(3x) + c2e^(-3x)sin(3x).

Next, we find the particular solution, yp(x), using the method of undetermined coefficients. Assuming a particular solution of the form yp(x) = Ae^(3x)cos(3x) + Be^(3x)sin(3x), we can find the values of A and B by substituting into the original equation and solving for the coefficients.