Differential Equations (Laplace Transforms)

Use Laplace Transforms to solve these IVP (partial fractions):

1. x''+4'+8x=e^-t x(0)=x'(0)=0

I don't know how to work with the imaginary roots.

2. x''''-x=0 x(0)=1 x'(0)=x''(0)=x'''(0)=0

3. x''''+13x''+36x=0 x(0)=x''(0)=0 x'(0)=2 x'''(0)=-13

Apply the convolution theorem to find the inverse Laplace transforms for:

4. F(s) = 1/(s(s^2+4))

5. F(s) = 1/(s(s^2+4s+5))

Apply the convolution theorem to derive the indicated solution x(t) of the given DE w/ initial conditions x(0)=x'(0) = 0

6.x''+4x'+13x=f(t);

x(t)=1/3 int[0,t] f(t-tau)e^-2(tau)*sin(3*tau) dtau

Solve the IVP

7. mx''+cx'+kx=f(t);

x(0) = x'(0)=0

m=1

k=4

c=5

f(t)=1 if 0=<t=<2

f(t)=0 if t>=2

Help on any is useful.