Soln space basis for all constant coeff homo linear DE's?

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1. Jan 6, 2016

kostoglotov

From what I've seen so far, the basis of the solution space for all the constant coefficient homo linear DE's have been linear combinations of the exponential function e or of some polynomial multiplied by the exponential function.

Is this always true that these DE's always result in solutions based on exponential, polynomial times exponential (and sin and cos combos)?

Is it ok to collapse all the sin and cos solutions into the space of exponential functions, since a complex component in the exponent gives cos and sin?

2. Jan 6, 2016

Staff: Mentor

Yes. Any constant coefficient, linear, homogeneous DE will look like this:
$a_{n}y^{(n)} + a_{n - 1}y^{(n - 1)} + \dots + a_1y' + a_0 = 0$
This results in a characteristic equation of $a_nr^n + a_{n - 1}r^{n - 1} + \dots + a_1r + a_0 = 0$
By the Fund. Thm. of Algebra, the above can be factored into linear and/or irreducible quadratic factors over R (or into linear factors over C).
Each linear factor r - a produces a solution of the form $e^{at}$. Each irreducible quadratic factor produces a pair of solutions of the form $e^{at}\cos(bt)$ and $e^{at}\sin(bt)$.
It's OK, but not usually done, since you will have e raised to complex powers.