Determine the Laplace transforms of the initial value problem (IVP)y′′−y′−30y=9u(t−5),y(0)=0,y′(0)=0𝑦″−𝑦′−30𝑦=9𝑢(𝑡−5),𝑦(0)=0,𝑦′(0)=0and obtain an expression for Y(s)=L(y)(t)𝑌(𝑠)=𝐿(𝑦)(𝑡). Do not find the inverse Laplace transform of the resulting equation.

The given initial value problem (IVP) is a second order linear differential equation with constant coefficients. The Laplace transform of the differential equation is given by:

L{y'' - y' - 30y} = L{9u(t-5)}

Taking the Laplace transform of each term separately, we get:

L{y''} - L{y'} - 30L{y} = 9L{u(t-5)}

Using the properties of Laplace transforms, we know that:

L{y''} = s^2Y(s) - sy(0) - y'(0) L{y'} = sY(s) - y(0) L{y} = Y(s) L{u(t-5)} = e^-5s/s

Substituting the initial conditions y(0) = 0 and y'(0) = 0, and the above expressions into the transformed differential equation, we get:

s^2Y(s) - 0 - 0 - sY(s) - 30Y(s) = 9e^-5s/s

Simplifying, we get:

(s^2 - s - 30)Y(s) = 9e^-5s/s

Finally, solving for Y(s), we get:

Y(s) = 9e^-5s / s(s^2 - s - 30)

This is the Laplace transform of the solution to the given initial value problem.