💬 👋 We’re always here. Join our Discord to connect with other students 24/7, any time, night or day.Join Here!

Like

Report

Get the answer to your homework problem.

Try Numerade Free for 7 Days

Verify that the given function is a solution to the given differential equation $\left(c_{1} \text { and } c_{2}\right.$ are arbitrary constants), and state the maximum interval over which the solution is valid.$$y(x)=c_{1} e^{x}+c_{2} e^{-2 x}, \quad y^{\prime \prime}+y^{\prime}-2 y=0$$.

is the solution

Calculus 2 / BC

Chapter 1

First-Order Differential Equations

Section 2

Basic Ideas and Terminology

Differential Equations

University of Michigan - Ann Arbor

Idaho State University

Boston College

Lectures

01:11

In mathematics, integratio…

06:55

In grammar, determiners ar…

06:51

Verify that the given func…

02:28

04:57

04:42

03:25

07:35

07:26

11:35

04:43

05:16

in this problem. We've been given a differential equation here and some why. And we want to check that this y of X is a solution to our differential equation. So to do that, first we check that wise at least twice differentials. Since we have wide double prime here, Well, it is because it's a function of, um, no addition and even the X, which it's gonna be infinitely differential so we can go ahead and take these derivatives Why Prime of X is equal Teoh C one e to the X plus. We're gonna do the chain role here minus two C two e to the minus two X and why Double prime of X is going to be equal to C one e to the X plus. We're gonna do the chain role again. Negative. Two times negative two is four C two e to the negative two X. Now that we have these, we want to plug them into our differential equation and make sure we get zero. So we're gonna take this wide double prime and plug it in here and the UAE prime and plug it in here and why of X plugged in to here, So let's go ahead and do that. Why? Double prime is C one e to the X plus four C to eat the negative two x plus Why Prime is going to be C one e to the X minus two C two e to the minus two x and finally minus two. Why is C one e to the X plus C two e to the negative two x So we take this and we can go ahead and distribute this minus two and we'll say this part over here is equal to negative two c one e to the X minus two C two e to the minus two x and we get some cancellations. Now, um, we have here foresee to eat the negative two x Well, that's gonna cancel with um minus two c two e to the negative two x in this minus C to eat of the up Sorry, minus two c two e to the negative two x thes two will cancel with that one. Um, then we have see one each of the X plus C one e to the X is two c one e to the X Well, that'll cancel with negative to see one e to the X. So we get This whole equation is equal to zero. And we wonder what interval that's true was true for X in any of the real numbers, from negative infinity to infinity. That's going to be true because you can plug in any X to the function each the X. That's no problem.

Numerade Educator

In mathematics, integration is one of the two main operations in calculus, w…

In grammar, determiners are a class of words that are used in front of nouns…

Verify that the given function is a solution to the given differential equat…