How do I find this inverse Laplace transform?

Question

ODE: $y'''' + 2y'' + y = 0$

Initial Data: $y(0) = 1, y'(0) = -1, y''(0) = 0, y'''(0) = 2$

$$Y(s^4 + 2s^2 + 1) = s^3 - s^2 + 2s$$

$$y = L^{-1} \left( \frac{s^3 - s^2 + 2s}{s^4 + 2s^2 +1} \right)$$

What do I have to do to solve this problem?

Answer 1

Hint: Perform a partial fraction decomposition and you find that

$$\frac{s^3 - s^2 + 2s}{s^4 + 2s^2 +1} = \frac{s^3 - s^2 + 2s}{(s^2 + 1)^2} = \frac{s-1}{s^2 + 1} + \frac{s+1}{(s^2 + 1)^2}$$

From there it's as simple as using your properties to find the inverse transform of each summand.