# Math Help - Piecewise Laplace 1

1. ## Piecewise Laplace 1

y'' + 81y = T(t)

where T(t) =

4t from 0 to 7
28 from 7 to infinity

y(0) = 0, y'(0) = 0

Find the equation you get by taking the Laplace transform of the differential equation and solve for Y(s)

I took Laplace of both side and answer comes out to be

(1/(1-e^(-7s)))((4(1-e^(-7s)(7s+1))/s^2+(28*e^(-7s)/s)))(1/(s^2+81))

2. Originally Posted by Phyxius117
y'' + 81y = T(t)

where T(t) =

4t from 0 to 7
28 from 7 to infinity

y(0) = 0, y'(0) = 0

Find the equation you get by taking the Laplace transform of the differential equation and solve for Y(s)

I took Laplace of both side and answer comes out to be

(1/(1-e^(-7s)))((4(1-e^(-7s)(7s+1))/s^2+(28*e^(-7s)/s)))(1/(s^2+81))

T(t) = 4t [H(t) - H(t - 7)] + H(t - 7) = 4t H(t) - (4t - 1) H(t - 7) where H(t) is the Heaviside Step Function: Heaviside step function - Wikipedia, the free encyclopedia

3. but my professor never taught us how to take Laplace of Heaviside functions...this is so frustrating lol

4. Thanks! I figured this one out. the answer is

(4(1-e^(-7s)(7s+1))/s^2+(28*e^(-7s)/s))(1/(s^2+81))