Refer a friend and earn $50 when they subscribe to an annual planRefer Now

Get the answer to your homework problem.

Try Numerade Free for 30 Days

Like

Report

Verify that $y(t)=A \cos (\omega t-\phi)$ is a solution to the differential equation $(1.1 .21),$ where $A$ and $\omega$ are nonzero constants. Determine the constants $A$ and $\phi$ (with $|\phi|<\pi$ radians) in the particular case when the initial conditions are$$y(0)=a, \quad \frac{d y}{d t}(0)=0$$

$yp(t)=\frac{F_{o}}{2_2o} t \sin (w t), w=w_{0}$

Calculus 2 / BC

Chapter 1

First-Order Differential Equations

Section 1

Differential Equations Everywhere

Differential Equations

Campbell University

Baylor University

University of Nottingham

Boston College

Lectures

01:11

In mathematics, integratio…

06:55

In grammar, determiners ar…

08:35

For all problems below, us…

03:37

Verify that $ y = -t \cos …

01:53

Find the particular soluti…

02:42

Verify that the given func…

02:00

Verify that the following …

05:53

Use the Laplace transform …

00:49

Find both the general solu…

01:19

Find the solution of the d…

00:52

01:05

Verify that $y=-t \cos t-t…

Okay, so just start this problem. We first are going to take a look at the roots of the exhilarate equation here, which is going to be p of our is equal to r squared. Plus, Omega, not squared, is equal to zero. So we're gonna have that r is equal to plus or minus I Omega. Uh, Omega, Not now. This part here has the route b is equal to omega eso if b is able to omega, Uh, but we need to take a look at the or compare the roots here so we'll have plus or minus i omega. So if omega is not equal to Omega, not so let's do that case first. Omega is not equal to Omega. Not so. If this is not Omega, not then our zp of tea is going to be equal to than just a not e to the Omega or Iomega like so. So this is going to have the derivative c p of t is equal to I unless should be Iomega t. So we'll have omega, uh, I or Iomega that a not e to the i omega t the men zP double prime of T is going to be able to the negative omega squared a not e to the I omega t like So let's plug these into our ah differential equation, which is going to solve our plus omega, not squared, is equal to is equal to. And then we're gonna have, um, on the right hand side, this is equal to f not. And then then e to the I omega t like so So Ah, oops, sorry. And then there should also be a zp here. So if we have negative omega squared Ah, so omega squared, um, a not e to the i Omega t and then plus omega not squared a not e to the I Omega T is equal to f not e to the I omega t Then we have omega not squared minus omega squared. A not is equal to f Not so a not is going to be able to f not over omega not squared minus omega squared. So we're gonna have our particular are complex valued particular solutions. E p of t is equal to f not over. Oh, Megan. Not square in minus omega squared. And then times e to the I Omega t so either the I omega t remember, we're gonna replace that, actually with co sign Omega T plus I sine omega t. Since we have the co sign part, we want the real part. So that's going to just be this here. So r y a p of tea is going to be equal to f not mining or divided by omega not squared minus omega squared cosigned omega t And this is for Omega not equal to Omega. Not so. That's for that scenario. Now we need toe Take a look at the case where omega is equal to Omega. Not so for Omega, equal to Omega. Not then That means our particular solution CP of tea. Then it's going to become a not tea e to the I Omega t like So So then zp prime t. Okay, so let's do that. So we're going tohave first times derivative. First time second is gonna be, uh, that and then plus, now we're going to have Iomega tea or so Iomega Times a not ah t e to the I Omega t. So, um, that is here. We took derivative of first time Second, then here we have derivative of second, which is the Iomega E to the Omega T times first, which is Iomega I not or a not tea. So the C double prime of tea this is equal to And then we have a not, uh, or sorry, i omega a not e to the i Omega t and then here we're gonna Ah, the derivative of this is just this. And then we multiply by Iomega. So we have plus Iomega a not e to the I Omega T and then plus I omega squared. That's going to become negative. Omega squared. Sorry. Negative. Omega squared. Okay, negative. Omega squared times a not and then ah t e to the I Omega t. Okay. Ah, we can now combine these two terms into just to be to Iomega a not e to the I Omega t and then minus We have our omega squared a not t e to the i Omega T. Next, we plug it into the complex value difference equation, which is E p uh, double prime plus. Sorry. Plus, and then Omega not is able to omega, so I'm just gonna write it as omega squared, and then zp is equal to R F not and then e to the i Omega t. So now plugging these into our ah, to this here we have to Iomega a not e to the i omega T minus omega squared a not t e to the i Omega T and then plus omega squared a not t e to the i omega T This is evil to f not e to the i omega t the these tube can now cancel out and we just have to Iomega a not it's gonna be able to f not when we also cancel out these terms here. So a not is kind of eagle two f not divided by two Iomega or this is also equal to F Um, sorry, I ah, negative. I f not over to omega. So zp of t is equal to, um negative I f not over to omega times t eats the Iomega t So this e to the Iomega's he I can replace with cosigned omega t plus I sine omega t And again, Since we're still taking the coastline part, we want the real part of this. So the real part of this is when we multiply this by this here. So this negative I becomes ice negative. I squared, which is positive. So why P of tea is gonna end up becoming okay. We have f not divided by to omega and then tea sign Omega t like so. And this is for Omega equals omega, not It's our final solution here.

View More Answers From This Book

Find Another Textbook

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…

For all problems below, use a complex-valued trial solution to determine a p…

Verify that $ y = -t \cos t - t $ is a solution of the initial value problem…

Find the particular solution to the differential equation $\frac{d u}{d t}=\…

Verify that the given function is a solution of the differential equation th…

Verify that the following functions are solutions to the given differential …

Use the Laplace transform to solve the initial-value problem $$\begin{aligne…

Find both the general solution of the differential equation and the solution…

Find the solution of the differential equation that satisfies the given init…

Verify that the given function y is a solution of the differential equation …

Verify that $y=-t \cos t-t$ is a solution of the initial-value problem$$…

16:03

A tank initially contains $w$ liters of a solution in which is dissolved $A_…

01:50

Find the general solution to the given differential equation and the maximum…

03:14

Determine an integrating factor for the given differential equation, and hen…

06:29

Use the modified Euler method with the specified step size to determine the …

01:26

Prove that the general solution to $y^{\prime \prime}-y=0$ on any interval $…

06:36

Solve the given initial-value problem.$$y^{\prime}=y^{3} \sin x, \quad y…

12:02

Use Euler's method with the specified step size to determine the soluti…

13:58

Solve the given differential equation.$$x y^{\prime}-y=\sqrt{9 x^{2}+y^{…

02:53

Use the technique derived in the previous problem to solve the given differe…

08:07

One solution to the initial-value problem$$\frac{d y}{d x}=\frac{2}{…

92% of Numerade students report better grades.

Try Numerade Free for 30 Days. You can cancel at any time.

Annual

0.00/mo 0.00/mo

Billed annually at 0.00/yr after free trial

Monthly

0.00/mo

Billed monthly at 0.00/mo after free trial

Earn better grades with our study tools:

Textbooks

Video lessons matched directly to the problems in your textbooks.

Ask a Question

Can't find a question? Ask our 30,000+ educators for help.

Courses

Watch full-length courses, covering key principles and concepts.

AI Tutor

Receive weekly guidance from the world’s first A.I. Tutor, Ace.

30 day free trial, then pay 0.00/month

30 day free trial, then pay 0.00/year

You can cancel anytime

OR PAY WITH

Your subscription has started!

The number 2 is also the smallest & first prime number (since every other even number is divisible by two).

If you write pi (to the first two decimal places of 3.14) backwards, in big, block letters it actually reads "PIE".

Receive weekly guidance from the world's first A.I. Tutor, Ace.

Mount Everest weighs an estimated 357 trillion pounds

Snapshot a problem with the Numerade app, and we'll give you the video solution.

A cheetah can run up to 76 miles per hour, and can go from 0 to 60 miles per hour in less than three seconds.

Back in a jiffy? You'd better be fast! A "jiffy" is an actual length of time, equal to about 1/100th of a second.