Question: In most explanations, you are given something like 

$ay''+by'+cy=0$

and the method starts with something like, "I can see that the exponential function would be a solution, and also adding two exponential functions together would work, so..." and so on.

What if I can't just magically see that this solution covers every possible solution? **How do you find a solution for this kind of equation algebraically?** I thought I had come up with a solution in an earlier question (<https://math.stackexchange.com/a/200082/21050>), but I didn't understand why my solution worked so that's no good.

I just learned a clever way to solve these when $c=0$, but assume that $c$ is not zero for your answer, please.

edit: I want to alter this question slightly: I can go ahead and assume that $e^{mx}$ is a solution and work from there. But what if I was braindead and couldn't make that guess? **How can I get $e^{mx}$ to spit out somewhere?**

Answer: 
For Excel, let $r$ be Pearson's r, $s\_x$ the standard deviation and $\bar{X}\_x$ the mean of all the numbers on the x-axis, $s\_y$ the standard deviation and $\bar{X}\_y$ the mean of all the numbers on the y-axis.

Then the slope will be $a = r \frac{s\_y}{s\_x}$ and y-intercept $b = \bar{X}\_y - a \bar{X}\_x$, for linear trendline $y = ax + b$.
When I have used the data analysis package for simple linear regression I always get a scatterplot of the data and have an option to add a trendline and also an option for the equation and R square to be printed on the graph. these days this can easily be done with the chart wizard.