Elementary Function Sampler Platter

Here is a sampling of the so-called Elementary Functions, and why you need to know about these in order to solve differential equations.

Polynomials

Exponentials

Sines and Cosines

A cool way to relate polynomials, exponentials, and sinusoids with the same DE!!

Based on the previous results about polynomials, exponentials, and sinusoids, we can deduce that they are solutions to a family of differential equations of a very simple type:

$\displaystyle f''(x)=kf(x)$

If $k>0$ , $f(x)$ is an exponential.

If $k=0$ , $f(x)$ is a first degree polynomial.

If $k<0$ , $f(x)$ is a sinusoid.

This suggests some sort of deep relationship between these three seemingly completely different kinds of functions.

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