Final answer:
To solve the given initial value problem, assume a solution of the form y = e^(rt), substitute it into the differential equation, find the characteristic equation, and solve for the solutions of y. Use the initial conditions to find the specific values of the constants, and combine the solutions to obtain the final solution.
Step-by-step explanation:
To solve the initial value problem y'' + 6y' + 9y = 0 with initial conditions y(0) = 2 and y'(0) = -7, we can assume a solution of the form y = e^(rt). Substitute this solution into the differential equation and solve for r. The characteristic equation is then obtained, which gives us the solutions for y.
After finding the solutions, we can use the initial conditions to find the specific values of the constants. The final solution is then obtained by combining the solutions and their corresponding constants. In this case, the solution is y(t) = 2e^(-3t) - 5e^(3t).