Question

Use the Laplace transform to solve the given initial value problem. y′′ −8y′ −128y=0;y(0)=7,y′ (0)=64 y(t)=

Answer 1

To solve the given initial value problem using the Laplace transform, we can find the solution for Y(s) in the time domain by using partial fraction decomposition and inverse Laplace transform.

Step-by-step explanation:

To solve the given initial value problem, we can use the Laplace transform. Applying the Laplace transform to the differential equation, we get

s^2Y(s) - 7s - 64 + 8(sY(s) - 7) - 128Y(s) = 0

Simplifying this equation, we obtain

Y(s) = \frac{71s + 64}{s^2 - 8s - 128}

Using partial fraction decomposition and inverse Laplace transform, we can find the solution for Y(s) in the time domain. The final answer is y(t) = 5e^(4t) - 12e^(-16t).