Using Laplace transform method, solve $y'''- 3y'' + y' - y = t^2e^{2t}$

Question

Using Laplace transform method, solve $$\dfrac {d^3y}{dt^3} - 3\dfrac {d^2y}{dt^2} + \dfrac {dy}{dt} - y = t^2e^{2t}$$ given $y (0) = 1, y′(0) = 0, y′′(0) = –2$.

I'm not able to factorize once the differential equation is formed.Can someone help me out? I was able to solve till here

Answer 1

Note that the Laplace transform of the LHS is given by:

$$s^3Y(s)-s^2y(0)-sy'(0)-y''(0)-3(s^2Y(s)-sy(0)-y'(0))+$$ $$sY(s)-y(0)-Y(s)\tag1$$

And for the RHS:

$$\frac{2}{(s-2)^3}\tag2$$

Answer 2

You have a problem with factorization of the polynomial on LHS: $$F(s)(s^3-3s^2+s-1)=\dfrac 2 {(s-2)^3}+s^2-3s-1$$ Since $s^3-3s^2+s-1$ has two complex roots and a real root that is not an integer. Are you sure the DE is correct and the inital conditions are correct ?

WA gives a very complicated answer for the homogeneous DE.