Suppose is a twice-differentiable real-valued function defined on some open subset of that satisfies the differential equation

where . To find an explicit equation for , we can invoke the following trick: suppose for some . Then

and so

Since the exponential function is never zero, we can divide by and obtain the **characteristic equation**

Now, we will use the following lemma:

Lemma 1If , then the equation has the two complex solutions

*Proof:* Dividing both sides by , we have . Now, as

we have

so that

Taking the square root of both sides, we obtain

and hence

the familiar “quadratic formula.”

From this lemma, we have that

For an example, consider the differential equation . Then the characteristic equation is

which has solutions , . So both and are solutions, as

and

As differentiation is a linear operator, are solutions of the ODE for any .