So, here is the extract from the reference handbook.

And it starts at some special cases, if the functions of x which

are the multipliers of the derivatives of the left hand side, are simple constants.

In other words, numbers or coefficients then the equation reduces to this,

bm, dny, dx to the m plus etcetera is equal to some function of x.

And furthermore if we have a homogeneous equation with constant coefficients,

in other words all those values b are simple numbers, simple constants.

Then we arrive at this particular equation, bn, dn,

d3ny, dx to the n, etcetera is equal to 0, and

this is a linear homogenous equation with constant coefficients.

And this equation has a general solution as given here,

y of x is = C1e, raised to the power 1x, etcetera,

where all the c1, c2, are coefficients.

And the rn exponents

are the nth route of the characteristic equation formed in this way.

P of r is equal to bn r to the n, etcetera is equal to 0.

So, let me do illustrate that with by means of an example.