Final answer:
To find a particular solution for the given nonhomogeneous differential equation, we assume a solution in the form Ax + B for the polynomial term and Ce^-x for the exponential term. We then determine the constants A, B, and C by substituting the assumed solution and its derivatives back into the original equation and finally verify the solution by matching both sides of the equation.
Step-by-step explanation:
The task is to find a particular solution yp for the nonhomogeneous differential equation y'' + 4y' + 5y = 15x + 3e-x. To achieve this, we look for a solution that specifically addresses the nonhomogeneous part of the equation, which in this case, is 15x + 3e-x. For the polynomial part 15x, we assume a solution of the form Ax + B where A and B are constants to be determined. For the exponential part 3e-x, we guess a solution of the form Ce-x since the exponential e-x is not a solution to the corresponding homogeneous equation y'' + 4y' + 5y = 0.
Once we have our assumed solution for yp, we calculate its first and second derivatives and substitute them back into the original differential equation to solve for constants A, B, and C. After determining these constants, we can state that yp in the form Ax + B + Ce-x is our particular solution.
To check if we have found the correct particular solution, we take the first and second derivatives of yp and substitute them back into the given differential equation. If the left-hand side of the equation simplifies to match the right-hand side, then our particular solution is indeed correct. This verification process is essential to confirm the solution's validity.