Call Now to Set Up Tutoring:

(888) 888-0446

Differential Equations : Initial-Value Problems

Study concepts, example questions & explanations for Differential Equations

Create An Account

All Differential Equations Resources

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

Example Questions

Example Question #11 : Initial Value Problems

Solve the following initial value problem: y' = 10(y - t) , y(0) = 1 .

Possible Answers:

$y = t + \frac{9e^{10t}}{10}$

$y = t + 10e^{10t}$

$y = t + \frac{1}{10} + 10e^{10t}$

$y = t + \frac{1}{10}$

$y = t + \frac{1}{10} + \frac{9e^{10t}}{10}$

Correct answer:

$y = t + \frac{1}{10} + \frac{9e^{10t}}{10}$

Explanation:

This type of problem will require an integrating factor. First, let's get it into a better form: y' - 10y = -10t . Once we have an equation in the form y' + p(t)y = q(t) , we find $\mu = e^{\int p dt} = e^{\int -10 dt} = e^{-10t}$ (where the constant of integration is omitted because we only need one, arbitrary integrating factor).

Once we do this, we can see that $(\mu y)' = \mu y' + \mu'y = e^{-10t}y' - 10e^{-10t}y = e^{-10t}(y' - 10t) = e^{-10t}\cdot-10t$

This is simply due to product rule, and then at the end, substitution of the original equation. Thus, as we know that $(\mu y)' = \mu q$ , we can just integrate both sides to find y.

$e^{-10t}y = \int-10te^{-10t}dt$ . A quick application of integration by parts with u = -10t and $dv = e^{-10t}$ tells us that the right hand side is $te^{-10t} + \frac{e^{-10t}}{10}+C$ . Dividing both sides by mu, we are left with $y = t + \frac{1}{10} + \frac{C}{e^{-10t}}$ . Plugging in the initial condition, we find $1 = \frac{1}{10} + C$ giving us $C = \frac{9}{10}$ . Thus, we have a final answer of $y = t + \frac{1}{10} + \frac{9e^{10t}}{10}$ .

Report an Error

View Differential Equations Tutors

Vicky
Certified Tutor

Clemson University, Doctor of Science, Civil Engineering.

View Differential Equations Tutors

Matthew
Certified Tutor

The University of Tennessee at Chattanooga, Bachelor of Science, Mathematics. University of Kentucky, Master of Science, Math...

View Differential Equations Tutors

Charles
Certified Tutor

University of Illinois at Urbana-Champaign, Bachelor of Science, Mathematics and Computer Science.

All Differential Equations Resources

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

Popular Subjects

Physics Tutors in Washington DC, Math Tutors in Atlanta, Statistics Tutors in Phoenix, MCAT Tutors in Seattle, GMAT Tutors in Phoenix, Biology Tutors in Miami, Algebra Tutors in Dallas Fort Worth, French Tutors in San Francisco-Bay Area, MCAT Tutors in Boston, Computer Science Tutors in Washington DC

Popular Courses & Classes

ISEE Courses & Classes in Chicago, SAT Courses & Classes in Dallas Fort Worth, SAT Courses & Classes in Seattle, GRE Courses & Classes in Denver, ISEE Courses & Classes in San Francisco-Bay Area, ISEE Courses & Classes in Boston, Spanish Courses & Classes in Philadelphia, MCAT Courses & Classes in Houston, LSAT Courses & Classes in Philadelphia, ACT Courses & Classes in Seattle

Popular Test Prep

GRE Test Prep in San Francisco-Bay Area, SSAT Test Prep in San Diego, SAT Test Prep in Chicago, MCAT Test Prep in Dallas Fort Worth, LSAT Test Prep in Seattle, SAT Test Prep in Dallas Fort Worth, ISEE Test Prep in Los Angeles, GRE Test Prep in Miami, GMAT Test Prep in Philadelphia, GMAT Test Prep in San Diego

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
101 S. Hanley Rd, Suite 300
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)

HIGH SCHOOL

GRADUATE SCHOOL

K-8

Search 50+ Tests

math tutoring

science tutoring

foreign languages

elementary tutoring

other

Search 350+ Subjects

Differential Equations : Initial-Value Problems

Study concepts, example questions & explanations for Differential Equations

All Differential Equations Resources

Example Questions

Example Question #11 : Initial Value Problems

All Differential Equations Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: