# Differential equation: distinct, real roots

• accountkiller
In summary, for the given equation, the general solution can be found using the "distinct, real roots" theorem or the "repeated roots" theorem. The quadratic formula can be used to find the roots, but it must be recognized when there are repeated roots and the appropriate theorem must be used to find the general solution.
accountkiller

## 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 y1=er1x and y2=erxx, so y(x) = c1er1x + c2r2x is the general solution.
"repeated roots" theorem - "If our roots are r1=r2, then the general solution is y(x) = (c1+c2x)er1x.

## The Attempt at a Solution

9y'' - 12y' + 4y = 0
9r2 - 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) = c1er1x + c2r2x, 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?

## 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 y1=er1x and y2=erxx, so y(x) = c1er1x + c2r2x is the general solution.
"repeated roots" theorem - "If our roots are r1=r2, then the general solution is y(x) = (c1+c2x)er1x.

## The Attempt at a Solution

9y'' - 12y' + 4y = 0
9r2 - 12r + 4 = 0
Using the quadratic formula, I got r = 2/3.
If you factored the equation above, you would get 9(x - 2/3)(x - 2/3) = 0. There are two roots. Are they distinct?
I'm confused because I only have one r. For the "distinct, real roots" theorem, the general solution is y(x) = c1er1x + c2r2x, 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?

Oh. I wasn't sure how to factor it so I just used the quadratic formula. So in order to get a correct answer, in this case the "repeated roots" theorem, I can't use the quadratic formula but have to factor?

No, you can use the quadratic formula - you just need to recognize when you are getting repeated roots. In this case, with the characteristic equation being 9r^2 - 12r + 4 = 0, the quadratic formula gives
r = (12 +/- sqrt(144 - 144))/18, so you have r = 2/3 +/- 0. So there's a repeated root, which happens whenever the discriminant (the part in the radical) is zero.

Ah, I see. Thanks so much for your help! :)

## 1. What is a differential equation with distinct, real roots?

A differential equation with distinct, real roots is a type of differential equation where the solutions have two different real values. This means that the equation has two distinct solutions that are not repeated and are real numbers.

## 2. How can you determine if a differential equation has distinct, real roots?

To determine if a differential equation has distinct, real roots, you can solve the equation using standard methods, such as separation of variables or the method of undetermined coefficients. If the solutions obtained are two distinct real values, then the equation has distinct, real roots.

## 3. What is the significance of having distinct, real roots in a differential equation?

Having distinct, real roots in a differential equation means that the solutions are unique and can be easily identified. It also indicates that the equation has a well-behaved behavior and is not chaotic or unstable.

## 4. Can a differential equation have more than two distinct, real roots?

Yes, a differential equation can have more than two distinct, real roots. However, the number of distinct, real roots is limited by the order of the equation. For example, a second-order differential equation can have a maximum of two distinct, real roots.

## 5. Are there any real-life applications of differential equations with distinct, real roots?

Yes, differential equations with distinct, real roots have various real-life applications in fields such as physics, engineering, and economics. They can be used to model and predict the behavior of systems that have two distinct solutions, such as damped oscillators and population growth.

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