Call Now to Set Up Tutoring:

(888) 888-0446

Create an account to track your scores
and create your own practice tests:

Create an Account Create a Practice Test

All Differential Equations Resources

1 Diagnostic Test 29 Practice Tests Question of the Day Flashcards Learn by Concept

Want to review Differential Equations but don’t feel like sitting for a whole test at the moment? Varsity Tutors has you covered with thousands of different Differential Equations flashcards! Our Differential Equations flashcards allow you to practice with as few or as many questions as you like. Get some studying in now with our numerous Differential Equations flashcards.

Flashcards: Linear Equations

Solve the initial value problem y'' - 4y' + 8y = 0 for y(0) = 1 and y'(0) = 2

$y = 4e^{2t} - 2e^{-2t}$

$y = e^{2t}\cos(2t)$

$y = e^{2t}\sin(2t)$

$y = 2e^{2t}\cos(2t) - 2e^{2t}\sin(2t)$

$y = e^{2t}\cos(2t) - e^{2t}\sin(2t)$

Report an Error Copyright Notice

Flashcards: Linear Equations

Solve the initial value problem y'' - 4y' + 8y = 0 for y(0) = 1 and y'(0) = 2

Your answer:

$y = 4e^{2t} - 2e^{-2t}$

Your answer:

$y = e^{2t}\sin(2t)$

Your answer:

$y = 2e^{2t}\cos(2t) - 2e^{2t}\sin(2t)$

Your answer:

$y = e^{2t}\cos(2t) - e^{2t}\sin(2t)$

Correct answer:

$y = e^{2t}\cos(2t)$

Explanation:

This is a linear higher order differential equation. First, we need the characteristic equation, which is just obtained by turning the derivative orders into powers to get the following:

$\lambda^2 - 4\lambda + 8 = 0$ We then solve the characteristic equation and find that $(\lambda - 2)^2 + 4 = 0; \lambda = 2 \pm 2i$ (Use the quadratic formula if you'd like) This lets us know that the basis for the fundamental set of solutions to this problem (solutions to the homogeneous problem) contains $e^{2t}\sin(2t), e^{2t}\cos(2t)$ .

As the given problem was homogeneous, the solution is just a linear combination of these functions. Thus, $y = c_1e^{2t}\sin(2t) + c_2e^{2t}\cos(2t)$ . Plugging in our initial condition, we find that 1 = c_2 . To plug in the second initial condition, we take the derivative and find that $y' = 2c_1e^{2t}(\sin(2t) + \cos(2t)) + 2e^{2t}(\cos(2t) - sin(2t))$ . Plugging in the second initial condition yields 2 = 2c_1 + 2 . Solving this simple system of linear equations shows us that

c_1 = 0; c_2 = 1

Leaving us with a final answer of $y = e^{2t}\cos(2t)$

(Note, it would have been very simple to find the right answer just by taking derivatives and plugging in, but this is not overly helpful for non-multiple choice questions)

Flashcards: Linear Equations

Find the general solution to y''' - y'' + y' - y = 0 .

$y = c_2e^{t}\sin(2t) + c_3e^{t}\cos(2t)$

$y = c_1e^{-t} + c_2\sin(t) + c_3\cos(t)$

$y = e^{t}(c_1 + c_2\sin(2t) + c_3\cos(2t))$

$y = c_1e^{t} + c_2\sin(2t) + c_3\cos(2t)$

$y = c_1e^{t} + c_2\sin(t) + c_3\cos(t)$

Report an Error Copyright Notice

Flashcards: Linear Equations

Find the general solution to y''' - y'' + y' - y = 0 .

Your answer:

$y = c_2e^{t}\sin(2t) + c_3e^{t}\cos(2t)$

Your answer:

$y = c_1e^{-t} + c_2\sin(t) + c_3\cos(t)$

Your answer:

$y = e^{t}(c_1 + c_2\sin(2t) + c_3\cos(2t))$

Your answer:

$y = c_1e^{t} + c_2\sin(2t) + c_3\cos(2t)$

Correct answer:

$y = c_1e^{t} + c_2\sin(t) + c_3\cos(t)$

Explanation:

This is a linear higher order differential equation. First, we need the characteristic equation, which is just obtained by turning the derivative orders into powers to get the following:

$\lambda^3 - \lambda^2 + \lambda - 1 = 0$ To factor this, in this case we may use factoring by grouping. More generally, we may use horner's scheme/synthetic division to test possible roots. Here are both methods shown.

$\lambda^3 - \lambda^2 + \lambda - 1 = (\lambda^3 + \lambda) - (\lambda^2 + 1) = \lambda(\lambda^2 + 1) - (\lambda^2 +1) = (\lambda^2 + 1)(\lambda - 1)$

Alternatively, the rational root theorem suggests that we try -1 or 1 as a root of this equation. Using horner's scheme, we see

$1 \left |\begin{matrix}1 && -1 && 1 && -1\\ && 1 && 0 && 1 \\ \hline \end{matrix}\right.$

$1 \: \, \: \: \: \, \, \: \: \: 0 \: \: \, \, \: \: \: \, \: \:1 \: \: \, \, \: \: \: \, \: \: 0$

Which tells us the the polynomial factors into $(\lambda - 1)(\lambda^2 + 1)$ and that $\lambda = 1, \pm i$ . This means that the fundamental set of solutions is e^t, sin(t), cos(t)

As the given problem was homogeneous, the solution is just a linear combination of these functions. Thus, $y = c_1e^{t} + c_2\sin(t) + c_3\cos(t)$ . As this is not an initial value problem and just asks for the general solution, we are done.

DMCA Complaint

If you believe that content available by means of the Website (as defined in our Terms of Service) infringes one or more of your copyrights, please notify us by providing a written notice (“Infringement Notice”) containing the information described below to the designated agent listed below. If Varsity Tutors takes action in response to an Infringement Notice, it will make a good faith attempt to contact the party that made such content available by means of the most recent email address, if any, provided by such party to Varsity Tutors.

Your Infringement Notice may be forwarded to the party that made the content available or to third parties such as ChillingEffects.org.

Please be advised that you will be liable for damages (including costs and attorneys’ fees) if you materially misrepresent that a product or activity is infringing your copyrights. Thus, if you are not sure content located on or linked-to by the Website infringes your copyright, you should consider first contacting an attorney.

Please follow these steps to file a notice:

You must include the following:

A physical or electronic signature of the copyright owner or a person authorized to act on their behalf; An identification of the copyright claimed to have been infringed; A description of the nature and exact location of the content that you claim to infringe your copyright, in \ sufficient detail to permit Varsity Tutors to find and positively identify that content; for example we require a link to the specific question (not just the name of the question) that contains the content and a description of which specific portion of the question – an image, a link, the text, etc – your complaint refers to; Your name, address, telephone number and email address; and A statement by you: (a) that you believe in good faith that the use of the content that you claim to infringe your copyright is not authorized by law, or by the copyright owner or such owner’s agent; (b) that all of the information contained in your Infringement Notice is accurate, and (c) under penalty of perjury, that you are either the copyright owner or a person authorized to act on their behalf.

Send your complaint to our designated agent at:

Charles Cohn Varsity Tutors LLC
8001 Forsyth Blvd., Suite 1050
St. Louis, MO 63105

Or fill out the form below:

Do not fill in this field

Contact Information

* Full legal name

Company name

* Phone number

* Email address

Address

* Address1

Address2

* City

* State

* Zipcode

* Country

Complaint Details

* URL of copyrighted work (where your original material is located)

* Please describe the copyrighted work so that it may be easily identified

I am the owner, or an agent authorized to act on behalf of the owner of an exclusive right that is allegedly infringed.

I have a good faith belief that the use of the material in the manner complained of is not authorized by the copyright owner, its agent, or the law.

This notification is accurate.

I acknowledge that there may be adverse legal consequences for making false or bad faith allegations of copyright infringement by using this process.

* Signature (your digital signature is legally binding)

View Differential Equations Tutors

Aymen
Certified Tutor

McGill University, Bachelor, Mechanical Engineering.

View Differential Equations Tutors

Daniel
Certified Tutor

State University of New York at Buffalo, Doctorate (e.g., PhD, MD, JD, etc.), Physics.

View Differential Equations Tutors

Anthony
Certified Tutor

Merrimack College, Bachelor of Science, Mathematics.

Popular Subjects

Spanish Tutors in Chicago, Math Tutors in Miami, Computer Science Tutors in Atlanta, Computer Science Tutors in Miami, LSAT Tutors in Denver, French Tutors in Seattle, Biology Tutors in Miami, Chemistry Tutors in Houston, GMAT Tutors in Houston, SAT Tutors in Atlanta

Popular Courses & Classes

SSAT Courses & Classes in Los Angeles, SSAT Courses & Classes in Philadelphia, GMAT Courses & Classes in Phoenix, GMAT Courses & Classes in Houston, SAT Courses & Classes in Miami, ISEE Courses & Classes in Philadelphia, MCAT Courses & Classes in San Diego, Spanish Courses & Classes in Denver, ACT Courses & Classes in Seattle, LSAT Courses & Classes in Boston

Popular Test Prep

SSAT Test Prep in Philadelphia, GRE Test Prep in New York City, ISEE Test Prep in Philadelphia, MCAT Test Prep in Seattle, MCAT Test Prep in Atlanta, GRE Test Prep in Phoenix, SSAT Test Prep in Los Angeles, ISEE Test Prep in Los Angeles, ISEE Test Prep in Houston, LSAT Test Prep in San Diego

HIGH SCHOOL

GRADUATE SCHOOL

K-8

Search 50+ Tests

math tutoring

science tutoring

foreign languages

elementary tutoring

other

Search 350+ Subjects

Create an account to track your scores
and create your own practice tests:

All Differential Equations Resources

Flashcards: Linear Equations

Flashcards: Linear Equations

Flashcards: Linear Equations

Flashcards: Linear Equations

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback:

HIGH SCHOOL

GRADUATE SCHOOL

K-8

Search 50+ Tests

math tutoring

science tutoring

foreign languages

elementary tutoring

other

Search 350+ Subjects

Create an account to track your scores and create your own practice tests:

All Differential Equations Resources

Flashcards: Linear Equations

Flashcards: Linear Equations

Flashcards: Linear Equations

Flashcards: Linear Equations

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Create an account to track your scores
and create your own practice tests: