Solving Constant Differencial Equations

Somebody asked me if I can teach how to solve equations:

on

Where are any real numbers.

It really is easy!

Write the quadradic equation,

Since this is a quadradic equation.

Let , this is the *discriminant*.

There are three possibilities:

------------------------------

1) : In this case the equation has two real solutions. Call them . Then the solution to is given by . Where are **any** real numbers.

2) . In this case the equation has one real solution. Call it . Then the solution to is given by .

3) . In this case the equation has two complex solutions. Since are real, the polynomial is a polynomial in real coefficients and hence the complex solutions come as complex conjuages. Thus let to the two solutions. Then is the solution to .