Take the Laplace transform of the following initial value problem and solve

for Y(s)={y(t)}:

y″+6y′+17y=T(t)

y(0)=0,y′(0)=0

Where T(t)={

t,     0≤t<1/2

1−t,    1/2≤t<1,

T(t+1)=T(t)

Y(s)=?

2)Take the Laplace transform of the following initial value and solve for Y(s)={y(t)}:

y″+4y=R(t)

y(0)=0,y′(0)=0

Where R(t)=sin(πt),R(t+1)=R(t).

Y(s)=?

3)Take the Laplace transform of the following initial value and solve for Y(s)={y(t)}:

y″+1y=

{

sin(πt),   0≤t<1

0,   1≤t

y(0)=0,y′(0)=0
Y(s)=?

Hint: write the right hand side in terms of the Heaviside function. 
Now find the inverse transform to find y(t)=?

(Use step(t-c) for uc(t) .)

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