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Determine all values of the constant $r$ such that the given function solves the given differential equation.$$y(x)=e^{r x}, \quad y^{\prime \prime}-y^{\prime}-6 y=0$$.

$r_{1}=3, r_{2}=-2$

Calculus 2 / BC

Chapter 1

First-Order Differential Equations

Section 2

Basic Ideas and Terminology

Differential Equations

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for this problem were given a differential equation over here and a potential solution. And we want to decide what our makes. Ah, this solution ballot. So to do so let's go ahead and find what are y prime and are wide, double prime are so derivative of wine is our times Eat our X and the second derivative is R squared times e to the R X. Now we can substitute this wide double prime Why prime and why into our differential equation and see for what are this will become zero. So it's going to do that. Why double prime is r squared times e to the r X minus y prime minus are times e to the r X and minus six. Y is minus six e to the r X s We want Teoh. Find out when this is equal to zero we're solving for our so we can go ahead and factor out this evil Rx Eat are Oops x get R squared minus R minus six and we want to solve when that is equal to zero. This is a quadratic equation in here. So he is. The Rx will not be equal to zero for any are. So we need that this part in here is equal to zero. And it's a quadratic equation. We can factor it. Um, it factored. We need numbers that multiply a negative one and add to negative six. So we get it. Um, I'm sorry. Other way around, we want numbers that multiply too negative. Six adds a negative one week ar minus three times are minus two. He r X equals zero. And so this equation is going to be equal to zero when R is equal to two or when R is equal to three. So if R is equal to two or three, this y of X is a solution to our differential equation.

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