×

### Let's log you in.

or

Don't have a StudySoup account? Create one here!

×

### Create a StudySoup account

#### Be part of our community, it's free to join!

or

##### By creating an account you agree to StudySoup's terms and conditions and privacy policy

Already have a StudySoup account? Login here

## MATH-M343/S343 Section 3.1 Notes

by: Kathryn Brinser

3

0

3

# MATH-M343/S343 Section 3.1 Notes MATH-S343

Marketplace > Indiana University > Mathematics > MATH-S343 > MATH M343 S343 Section 3 1 Notes
Kathryn Brinser
IU
GPA 4.0

Get a free preview of these Notes, just enter your email below.

×
Unlock Preview

### Preview these materials now for free

Why put in your email? Get access to more of this material and other relevant free materials for your school

Covers homogeneous second-order differential equations.
COURSE
Honors Differential Equations
PROF.
Michael Jolly
TYPE
Class Notes
PAGES
3
WORDS
CONCEPTS
math-s343, math-m343
KARMA
25 ?

## Popular in Mathematics

This 3 page Class Notes was uploaded by Kathryn Brinser on Tuesday October 18, 2016. The Class Notes belongs to MATH-S343 at Indiana University taught by Michael Jolly in Fall 2016. Since its upload, it has received 3 views. For similar materials see Honors Differential Equations in Mathematics at Indiana University.

×

## Reviews for MATH-M343/S343 Section 3.1 Notes

×

×

### What is Karma?

#### You can buy or earn more Karma at anytime and redeem it for class notes, study guides, flashcards, and more!

Date Created: 10/18/16
S343 Section 3.1 Notes- Homogeneous Second Order Equations with Constant Coefficients 9-27-16 ???? ???? ????????  Second order ODE- takes form ????????2 = ????(????,????, ????????) with independent variable ???? (or ????) and dependent variable ???? o ODE linear if ????(????,????, ????????) = ???? ???? − ???? ???? ( )???????? − ???? ???? ???? (if ???? is linear in ???? and) ???????? ???????? ???????? o ????,????,???? are specified functions of ???? but do not depend on ???? ′′ ′  Second order linear equation- takes form ???? + ???? ???? ???? + ???? ???? ???? = ???? ???? ; rewritten form of ???? ???? ???????? ????????2 + ???? ???? ) ????????+ ???? ???? ???? = ???? ???? ( ) o Given ???? ???? ???? + ???? ???? ???? + ???? ???? ???? = ???? ???? , if ???? ???? ≠ 0, divide all terms by it to get correct form o Homogeneous if ???? ???? = 0, nonhomogeneous otherwise  Homogeneous with constant coefficients- ???????? + ???????? + ???????? = 0 ???????? o To solve, look for solution of form ???? = ???? (convenient form because derivatives repeat themselves with only a factor of ???? as a difference, so they can be factored out) ′ ????????  ???? = ????????  ???? = ???? ????????( ????????) = ???? ???? ????????  ???????? + ???????? + ???????? = ???????? ???? 2 ???????? + ???????????? ???????? + ???????? ????????= 0 ???????? 2  ???? (???????? + ???????? + ???? = 0  ???? ????????≠ 0 for all ????, so ???????? + ???????? + ???? = 0; known as characteristic equation o Find correct value(s) of ???? by finding roots of characteristic equation o 3 possibilities for roots1???? 2???? :  Real and distinct ???? 1 ???? 2)  Real and repeated ???? =1???? 2) ( )  Complex ???? = 1 + ????????,???? = ???? 2 ???????? o Solutions to equation: ???? 1 ???? ????1????, 2 = ???? ????2???? o For any ???? ,???? , general solution that also works is ???? = ???? ???? + ???? ???? = ???? ???? ????1???? + ???? ???? 2 ????(linear 1 2 1 1 2 2 1 2 combination of two known solutions), assuming ???? ,????1re2l and distinct  ???? = ???? ???? ???? ????1???? + ???? ???? ???? ????2???? ′′ 1 12 1 ???? 2 2 2 ????2????  ???? = ???? ???? 1 1 + ????2????2????  Substitute these into ???????? + ???????? + ???????? = 0: ′′ ′ 2 ????1???? 2 2 ???? ????1???? 2 ???? ????1???? ????2???? ???????? + ???????? + ???????? = ???? ???? ???? ????( 1 1 + ???? 2 2 ) + ???? ???? 1 1 + ????2 2???? ) + ???? ???? 1 + ???? 2 ) = ???????? 1 ????1 1 ????+ ???????? 2 2 2 ????+ ???????? 1 1 ????1???? + ????????1???? ????1????+ ???????? 2 ????2???? = ???? ???? 1 (???????? + ???????? + ???? + ???? ???? ????2????(???????? + ???????? + ???? ) 1 ???? ???? 1 ???? ???? 2 2 2 ′′ ′ = ???? 1 1 (0 + ???? ????2 2 (0) ????1,????2known to satisfy ???????? + ???????? + ???????? = 0 = 0 ′′ ′  ∴ ???? = ???? 1 1 + ???? ???? ????2 2 a solution of ???????? + ???????? + ???????? = 0  Ex. ???? − 5???? + 6???? = 0 2 o Characteristic: ???? − 5???? + 6 = 0 (???? − 3 ???? − 2 = 0 ???? = 2,3 2???? 3???? ∴ ???? 1 ???? ,???? = ????2 are both solutions o General solution: ???? = ???? ???? 2????+ ???? ???? 3???? ′ 1 2 o Let ???? 0 = 4, ???? 0 = −1 be initial values (must be at same ???? value):  ???? 0 = 4 = ???? ???? + ???? ???? 0 1 2 ????1+ ???? =24  ???? ???? =) ???? (???? ???? 2???? + ???? ???? 3????) ???????? 1 2 = ???? 1???? 2????) + ????23???? 3????) 2???? 3???? ′ = 2???? 1 + 3????2????0 0  ???? 0 = −1 = 2???? ???? + 1???? ???? 2 2???? + 3???? = −1 1 2  Solve system of equations: ????1+ ????2= 4 → ???? =14 − ???? 2 2 4 − ???? )+ 3???? = −1 2 2 8 − 2????2+ 3???? 2 −1 ????2+ 8 = −1 ???? = −1 − 8 = −9 2 ????1= 4 − −9 = 13  ∴ particular solution to IVP is ???? = 1− 9???? 3????  As ???? → ∞, negative coefficient in front of “faster” exponential dominates, ???? → −∞  Ex. Solve the initial value problem ???? + 4???? + ???? = 0, ???? 0 = 0, ???? 0 = −1. o ???? + 4???? + 1 = 0  ???? = −4±√16−4 1 1( = −4± √2 = −4±2 √ = −2 ± 3√ real and distinct 2 2 2  General solution: ???? =1???? ????2+ √ ) + ????2????(−2−√3)???? 0 0 o ???? 0 = 0 = ???? ????1+ ???? ???? 2 ????1+ ???? 2 0 ′ (2+ √ ) (2− √ ) o ???? ???? = −2(+ 3 ???? √ ) 1 + (2 − 3 √ ) 2 o ???? 0 = −1 = −2 ( 3 ???? ????√+ )21− 3 ????(???? √ ) 2 0 (−2 + 3√????)+1−2 ( 3 ???? =√−) 2 o Solve system: ????1+ ???? 2 0 → ???? =1−???? 2 (−2 + 3√−)( 2)+ −( − 3 √ =)−2 (2 − √ ????)+2−2(− 3 ???? √ −) 2 2???? − 3√ − 2???? − 3???? =√−1 2 2 2 2 −2 √???? =2−1 ???? = 1 = 1 2 2√3 √12 ???? = −1 1 √12 o ???? = −1 ????(−2+√3)????+ 1 ????(−2−√3)???? √12 √12 o As ???? → ∞, ???? → 0 (both exponents negative) ′′ ′ ′  Ex. Solve the initial value problem ???? + 2???? = 0, ???? 0 = 5, ???? 0 = 6. o ???? + 2???? = 0 ???? ???? + 2 = 0 ???? = −2,0 ???? = ????1????−2???? + ????2???? = ???? 1 −2????+ ????2 o ???? 0 = 5 = ???? ???? + ???? 1 2 ????1+ ???? 2 5 o ???? ???? = −2???? ???? 1 −2???? o ???? 0 = 6 = −2???? 1 o Solve system: ????1+ ???? 2 5 → ???? =25 − ???? 1 −2???? 1 6 → ???? =1−3 ????2= 5 − −3 = 8 o ???? = −3???? −2????+ 8 o As ???? → ∞, ???? → 8  Ex. Solve the initial value problem ???? + ???? − 2???? = 0, ???? 0 = ????, ???? 0 = 2. Then find ???? so that the solution approaches 0 as ???? → ∞. 2 o ???? + ???? − 2 = 0 (???? − 1 ???? + 2 = 0 ???? = −2,1 ???? = ????1???? + ???? 2 −2???? o ???? 0 = ???? = ???? + ???? ′ ????1 2 −2???? o ???? ???? = ???? 1 − 2???? ????2 o ???? 0 = 2 = ???? −12???? 2 o Solve system: ????1+ ???? 2 ???? → ???? =1???? − ???? 2 ???? − ???? − 2???? = 2 2 2 ???? − 3???? 2 2 −3???? 2 2 − ???? ????−2 ????2= 3 ????−2 3????−????+2 2????+2 ????1= ???? − 3 = 3 = 3 2????+2 ???? ????−2 −2???? o ???? = 3 ???? + 3 ???? 2????+2 o We want ????1= 0 (second term already goes to 0 due to negative exponent), s3 = 0 → ???? = −1 makes all solutions approach 0 as ???? → ∞ o When ???? ≠ −1, solution unbounded (???? → −∞ or ???? → ∞ depending on value of ????)

×

×

×

### You're already Subscribed!

Looks like you've already subscribed to StudySoup, you won't need to purchase another subscription to get this material. To access this material simply click 'View Full Document'

## Why people love StudySoup

Jim McGreen Ohio University

#### "Knowing I can count on the Elite Notetaker in my class allows me to focus on what the professor is saying instead of just scribbling notes the whole time and falling behind."

Kyle Maynard Purdue

#### "When you're taking detailed notes and trying to help everyone else out in the class, it really helps you learn and understand the material...plus I made \$280 on my first study guide!"

Bentley McCaw University of Florida

#### "I was shooting for a perfect 4.0 GPA this semester. Having StudySoup as a study aid was critical to helping me achieve my goal...and I nailed it!"

Parker Thompson 500 Startups

#### "It's a great way for students to improve their educational experience and it seemed like a product that everybody wants, so all the people participating are winning."

Become an Elite Notetaker and start selling your notes online!
×

### Refund Policy

#### STUDYSOUP CANCELLATION POLICY

All subscriptions to StudySoup are paid in full at the time of subscribing. To change your credit card information or to cancel your subscription, go to "Edit Settings". All credit card information will be available there. If you should decide to cancel your subscription, it will continue to be valid until the next payment period, as all payments for the current period were made in advance. For special circumstances, please email support@studysoup.com

#### STUDYSOUP REFUND POLICY

StudySoup has more than 1 million course-specific study resources to help students study smarter. If you’re having trouble finding what you’re looking for, our customer support team can help you find what you need! Feel free to contact them here: support@studysoup.com

Recurring Subscriptions: If you have canceled your recurring subscription on the day of renewal and have not downloaded any documents, you may request a refund by submitting an email to support@studysoup.com

Satisfaction Guarantee: If you’re not satisfied with your subscription, you can contact us for further help. Contact must be made within 3 business days of your subscription purchase and your refund request will be subject for review.

Please Note: Refunds can never be provided more than 30 days after the initial purchase date regardless of your activity on the site.