I have the following question. Find the general solution of:
$$y^{(5)} - 2y^{(4)} + 3y^{(3)} = 0$$
I found the characteristic equation
$$r^3(r^2 - 2r + 3) = 0$$
With roots: $$r = 0, \ \ multiplicity:3 \\and \ \ \ r = 1\pm\sqrt2i$$
The general solution I found is: $$y = C_1e^{(0)t}+C_2te^{(0)t}+C_3t^2e^{(0)t}+C_4e^tcos(\sqrt2t)+C_5e^tsin(\sqrt2t)$$
Simplifying to: $$y = C_1+C_2t+C_3t^2+C_4e^tcos(\sqrt2t)+C_5e^tsin(\sqrt2t)$$
Is this the correct solution? I am unsure whether or not root multiplicity applies to an $\mathbf{r^3 = 0}$ root. Thanks in advance for the help!