For homogenous linear differential equation with constant coefficients, the general solution has the form
the so called characteristic polynomial.
In case of repeated roots, appear associated polinomials in
It is found that the differential equation can be converted to a
polynomial equation, with the same coefficients, by the substitution
With D for the differentiation operator
It follows that , for every root
Also, an arbitrary scalar
Theoretically, reduction of the order of the differential equation by
every integration produces one constant of integration. So,
successive integration n times to produce the general solution
would deposit n constants of integration.
Now, the linear sum
And so, we are justified in stating that
There are particular cases like
leads to the degenerate case.
Here, 0 is a thrice repeated root. of the characteristic equation.
The part ax + b comes from direct integration, twice in succession,
for removing D^2. The other operator D+1 gives the part
Substitute separately both in the differential equation and see
how it works.