I have three questions out of 50 that I just cant seem to ge

I have three questions out of 50 that I just can\'t seem to get right.... Use the Laplace transform to solve the given initial-value problem.

a) y\"-6y\'+9y=t, y(0)=0, y\'(0)=1

b) y\"-6y\'+13y=0, y(0)=0, y\'(0)=-7

c) y\"-y\'=e^tcost, y(0)=0, y\'(0)=0

Solution

y\"-6y\'+9y=t, y(0)=0, y\'(0)=1

Apply the Laplace transform on the differential equaiton.

s^2*Y-sy(0)-y\'(0)-6[sY-y(0)]+9Y=1/s^2

s^2*Y-1-6sY+9Y=1/s^2

Y(s^2-6s+9)=1/s^2

Y=1/(s^2-6s+9)*s^2)

=1/(s^2(s-3)^2)

=2/27(s-3)+1/9(s-3)^2+2/27s+1/9s^2       (Use the partial fractions)

Apply the inverse laplace transform.

y(t)=-(2e^3t)/27+(e^3t)/9+2/27+t/9