- #1

accountkiller

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## Homework Statement

Find the general solution by using either the "distinct, real roots" theorem or the "repeated roots" theorem.

9y'' - 12y' + 4y = 0

## Homework Equations

"distinct, real roots" theorem - "If our roots are real & distinct, we should have solutions y

_{1}=e

^{r1x}and y

_{2}=e

^{rxx}, so y(x) = c

_{1}e

^{r1x}+ c

_{2}

^{r2x}is the general solution.

"repeated roots" theorem - "If our roots are r

_{1}=r

_{2}, then the general solution is y(x) = (c

_{1}+c

_{2}x)e

^{r1x}.

## The Attempt at a Solution

9y'' - 12y' + 4y = 0

9r

^{2}- 12r + 4 = 0

Using the quadratic formula, I got r = 2/3.

I'm confused because I only have one r. For the "distinct, real roots" theorem, the general solution is y(x) = c

_{1}e

^{r1x}+ c

_{2}

^{r2x}, so what do I do with just my one r? Do I just leave out the second part of the general solution? Or do I use the "repeated roots" theorem since I only have one r and that general solution only includes one r?