Question

Proceed as in this example to find the solution of the given initial-value problem. Evaluate the integral that defines yp(x). y'' - 9y = e^(3x), y(0) = 0, y'(0) = 0.

a) Missing information
b) (1/12)e^(3x)
c) (1/9)e^(3x)
d) (1/6)e^(3x)

Answer 1

To find the solution of the given initial-value problem, proceed by evaluating the integral that defines yp(x). The correct solution is (1/12)e^(3x).Option B is the correct answer.

Step-by-step explanation:

To find the solution of the given initial-value problem, we need to evaluate the integral that defines yp(x). The equation is y'' - 9y = e^(3x), with initial conditions y(0) = 0 and y'(0) = 0. Here's how to proceed:

1. Identify the differential equation as a second-order linear homogeneous equation with constant coefficients.
2. Find the complementary solution (yc(x)) by solving the associated homogeneous equation, which is y'' - 9y = 0.
3. Find the particular solution (yp(x)) by using a method like undetermined coefficients or variation of parameters. In this case, the non-homogeneous term e^(3x) suggests trying a particular solution of the form yp(x) = Ae^(3x), where A is a constant to be determined.
4. Substitute the particular solution into the differential equation and solve for the constant A.
5. Add the complementary solution and the particular solution to get the general solution (y(x) = yc(x) + yp(x)).
6. Apply the initial conditions to find the values of the constants and obtain the specific solution.

After evaluating the integral to find yp(x), the correct option is (b) (1/12)e^(3x).