Characteristic polynomial help

[roryhand]

y^(7)-y^(6)-2y^(4)+2y^(3)+dy-y=0

Note: There is exactly one real zero of the characteristic polynomial and it
has multiplicity 3 (it is a positive integer!). The other zeros are complex
and they have multiplicity 2.

Sadly I missed this lecture day, and am unsure of where to start. Any differential equation demi-gods out there?

 

[Pyrrhus]

For an equation of order n, if a root (say r1) has a multiplicity s (s =< n), where x is the independent variable

$$e^{r_{1}x}, xe^{r_{1}x}, x^{2} e^{r_{1}x}, ..., x^{s-1} e^{r_{1}x}$$

For complex roots, let's say $a+bi$ is repeated s times, then the complex conjugate $a-bi$ is also repeated s times, therefore the solutions for real valued functions, where x is the independent variable:

$$e^{ax} \cos{bx}, e^{ax} \sin{bx}, xe^{ax} \cos{bx}, xe^{ax} \sin{bx},..., x^{s-1} e^{ax} \cos{bx}, x^{s-1} e^{ax} \sin{bx}$$