# Question #76309

##### 2 Answers

For homogenous linear differential equation with constant coefficients, the general solution has the form

the so called characteristic polynomial.

If

In case of repeated roots, appear associated polinomials in

The constants

See explanation.

#### Explanation:

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

gives.

As

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,

Here, 0 is a thrice repeated root. of the characteristic equation.

For

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.

.