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Let us have a look at equations of the form

(This, by the way, is also called a homogenous equation.) Except for the first term it is has the same form as

We can rewrite that as

Integrating this gives us

or

The sign can be absorbed into the constant, finally giving us

So it may be an idea to test if the same type of solution could be solving our initial equations. Let us assume we have a solution of the form

We then have

Substituting this into our initial equation gives us

This could be factorized as

So if *r* have a value such that the second factor is 0 then we have a solution. The equation

is called the characteristic equation, and the solutions of our original equation will depend on the solutions to the characteristic equation.

**An example**

Say we have

That gives the characteristic equation

We can solve that by (for example) splitting -15 into two factors that add up to 2. Doing so we get

with the solutions *r*=3 and *r*=-5. We will thus have as solutions to our differential equation

where C and D are constants. Matter a fact, as you can quite easily verify, we have

as a solution (matter a fact it is the general solution). But a quadratic equation, in this case the characteristic equation) may have two real roots, two equal real roots or two complex roots. So what does these different cases correspond to? Let us have a look in the following few pages.

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