Final answer:
In order to solve for the response of the second-order differential equation with the given initial conditions, we first find the general solution using the characteristic equation, then apply those conditions to determine the constants and find the specific response.
Step-by-step explanation:
Response to a Second-Order Differential Equation
To obtain the response of the given equation x´ + 4xˆ + 8x = 0 with initial conditions x(0) = 0 and xˆ(0) = 1, we must solve the second-order differential equation. We have the known initial conditions x(0) = 0 and xˆ(0) = 1. The best approach is to find a general solution for the homogeneous equation and then apply the initial conditions to find the particular solution.
First, we solve the characteristic equation associated with the given differential equation, which is a quadratic equation: λ² + 4λ + 8 = 0. Using the quadratic formula, we find the roots of the characteristic equation.
Once we have the general solution, we apply the initial conditions to determine the constants in the solution and thus find the specific response of the system.
The characteristic equation might have real or complex roots, and depending on the nature of these roots, the general solution will involve exponential functions, sine and cosine functions, or a combination of both.
After determining the general solution, we use the initial conditions x(0) = 0 and xˆ(0) = 1 to solve for the constants in the solution. This process involves substituting the initial conditions into the solution and solving the resulting system of equations.