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# Math 340 - Week 3 Math 340

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This 6 page Class Notes was uploaded by Susan Ossareh on Sunday January 31, 2016. The Class Notes belongs to Math 340 at Colorado State University taught by in Spring 2016. Since its upload, it has received 14 views. For similar materials see Intro-Ordinary Differen Equatn in Math at Colorado State University.

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Date Created: 01/31/16

Math 340 Lecture – Introduction to Ordinary Differential Equations – February 1, 2016 What We Covered: 1. Worksheet 3 – Linear Equations a. Highlights i. Find a general solution to the following linear equations. Find also a particular solution if an initial condition is given. As a side note: you should always use the integrating factor when dealing with linear equations. 1. (1 + ???? ???? + ???? = sin( ????) ′ ???????? 2. ???? = ???????? + ???????? a. Where c and m are constants ′ −???? 3. sin ???? ???? + cos ???? ???? = ???? ,????????????ℎ???? 0 = 1 a. Exam-like question 1 2???? 4. ???? = 2 + ???????????????? ???? ,????????????ℎ???? ???? = ???? 2 ???? ???? a. Exam-like question 2. Course Content – Chapter 2: First-Order Differential Equations a. Section 2.4: Linear Equations Continued i. Review 1. First-Order Linear Equations ′ ???? = ???? ???? ???? + ????(????) ????.???? = ???? ???? ???? ⟶ ???????????????????????????????????? **product of a function of f and y** ????????.????(????) ≠ 0 ii. Example 1: ???? ???? = −2???????? + ???? ???? 2 ′ ???? ???? ???? + 2???????? = ???? **Here we can make ???? ???? = ????????′ and 2???????? = ????′????, so according to the chain rule…** (???????? )′= ???? ???? 2 ′ ???? (???? ???? ) = ???? 2 ???? ???? ???? = ∫???? ???????? ???? ???? ???? = ???? + ???? ???? ( ) ???? + ???? ???? ???? = ????2 iii. Example 2: 1. The linear form is ???? = ???? ???? ???? + ???? ???? . Here is our given equation ′ (1 + ???? ???? + ???? = sin( ????) 2. So based off of the form and the equation we are given, we want y’ to have a coefficient of 1 so it matches the linear form. So we are going to divide by (1+x) ′ ???? sin ????) ???? + 1 + ???? = 1 + ???? 3. So now we have to solve for the integrating factor. To find the integrating factor, we use this formula: ???? − ∫???? ???? ???????? 1 ∫1+???? ???????? ln(1+????) ???? = ???? = 1 + ???? **When looking at natural log and the integration, we didn’t include absolute value bars or +C because we only want one solution** 4. Now that we have the integrating factor, we multiply it into our equation ( ) (1 + ???? (???? +′ ???? ) = (sin ???? )(1 + ????) 1 + ???? 1 + ???? (1 + ???? ???? + ???? = sin(????) 5. This gives us our original starting equation, however our work wasn’t futile, because it allows us to follow the chain rule… (1 + ???? ???? ]′ = sin ????) (1 + ???? ???? = ∫sin(????)???????? (1 + ???? ???? = −cos ???? + ???? 6. This leads us to our general solution… −cos( ????) ???? ???? ???? = + 1 + ???? 1 + ???? Suggested Homework: Become comfortable with the properties of natural log (there is a sheet on the class website with detailed information) Finish Worksheet 3 Section 2.4: 2, 3, 4, 8, 11, 13, 14, 16, 36, 38 Math 340 Lab – Introduction to Ordinary Differential Equations – February 2, 2016 What We Covered: 1. Well we actually didn’t cover anything 2. Surprise Snow Day! Yay!! 3. But there is still a quiz on Wednesday (tomorrow) a. It will be on separable and linear equations 4. Enjoy your snow day! a. Woot Math 340 Lecture – Introduction to Ordinary Differential Equations – February 3, 2016 What We Covered: 1. Quiz 03 a. Highlights i. Covered separable and linear equations 2. Course Content – Chapter 2: First-Order Equations a. Section 2.4: Linear Equations Continued i. Example 1: 1. So we start off with our linear equation that we can reformat to fit the linear form ′ ( ) ????????????????:???? = ???? ???? ???? + ????(????) sin ???? ???? + cos ???? ???? = ???? −???? cos(????) ???? −???? ???? + ???? = sin ????) sin(????) cos ???? ) ????−???? ???? = − ???? + sin ????) sin(????) −∫ ???? ???? ???????? 2. We then solve for the integrating factor using the formula: ???? ( ) ∫cos(????????) ????− ∫???? ???? ????????= ???? sin( ????) **We can solve the integral using u-substitution** ???? = sin ???? ???????? = cos ???? ???????? ???????? ????∫???? = ???? ln|????= ???? = sin( ????) 3. After getting the integrating factor, we can multiply it into the equation and come to the general equation −???? ′ cos ???? ) ???? (sin ???? (???? + ????) = ( )((sin ???? ) sin ????) sin ???? ) [sin ???? ???? ]′= ???? −???? sin ???? ???? = ∫???? −???????????? −???? sin ???? ???? = −???? + ???? −???? −???? + ???? ???? ???? = sin ????) 4. We can then solve for the particular solution using the initial condition ???? ( ) = 1. 2 −???? ???? −???? 2 + ???? ???? ( ) = ???? = 1 2 sin ) ???? 2 −???? −2 + ???? = 1 1 −???? ???? = 1 + ???? 2 5. We then have the solution −???? −????2 −???? + 1 + ???? ???? ???? = sin ????) ii. So let’s break down the steps for solving linear equations! Yay! 1. First set up the given equation to fit the linear form ???? = ???? ???? ???? + ????(????) 2. Solve for the integrating factor ????− ∫???? ???? ???????? 3. Multiply the integrating factor into the equation 4. Integrate the equation 5. Solve for x iii. Example 2: So let’s practice again applying these steps, shall we? 1. First, set up the given equation to fit the linear form 1 2???? ???? = + ????????????????(????) ???? ????2 ????′ 2???? ???? ( − ) = (???????????????? ???? )???? ???? ????2 ′ 2???? 2 ???? − ???? = ???? cos(????) 2. Solve for the integrating factor −∫ ???? ???? ???????? − ∫2???????? −2ln ????) −2 ???? = ???? ???? − ???? = ???? 3. Multiply the integrating factor into the whole equation −2 ′ 2???? 2 −2 ???? (???? − ) = (???? cos ???? )???? −2???? ′ (???? ????) = cos(????) 4. Integrate the equation ????−2???? = ∫cos ???? ????????) ????−2 ???? = sin ???? + ???? 2 ???? ???? = ???? (sin ???? + ????) 5. Solve for x given the initial condition ???? ???? = ???? 2 iv. Example 3: Aaaand for further practice, let’s do this one more time 1. First, set up the given equation to fit the linear form ′ ???? = tan ???? ???? + sin( ????) ???? − tan ???? ???? = sin( ????) 2. Solve for the integrating factor ????− ∫???? ???? ????????= ???? −∫ tan ???? ????= ???? −(−ln cos ????))= ???? ln cos ????)= cos( ????) 3. Multiply the integrating factor into the whole equation (cos ???? (???? − ???????????????? ???? ) = (sin ???? )(cos ???? ) ( ) [cos ???? ???? ]′= sin ???? cos( ????) 4. Integrate the equation cos ???? ???? = ∫sin ???? cos ???? ???????? ) **Here we can use u-substitution to solve the equation** ???? = sin ???? ???????? = cos ???? ???????? 2 2 ???? ???????????? (????) ∫???????????? = + ???? = + ???? 2 2 2 ???????????? (????) cos ???? ???? = + ???? 2 5. Solve for x ???????????? (????) ???? ???? ???? = + 2cos(????) cos( ????) Suggested Homework: Section 2.4: 2, 3, 4, 8, 11, 13, 14, 16, 36, 38 Math 340 Lecture – Introduction to Ordinary Differential Equations – February 5, 2016 What We Covered: 1. Worksheet 4 a. Highlights i. Prove the equation 2???? − 1 ???????? + 3???? + 7 ???????? is exact and find an explicitly defined solution ii. Find the values of k such that the following equation is exact: 2???? − ???????????????? ???????? +( ) ???????? 4)???????? − (20???? ???????? + ???????????????? ???????? )???????? = 0 2 iii. Consider the equation ???????????? + ???????? − ???? ???????? = 0) 1. Verify that the equation is not exact 1 2. Multiply the equation by ???? ???? =) 2. Verify that the resulting equation is ???? exact and solve the equation. This means that u(x) is an integrating factor for the equation 4????2 iv. Prove the ???? ???? = ???????? is an integrating factor of 2???? − 6???? ???????? + (3???? − ) = 0 and ???? find the general solution 2. Course Content – Chapter 2: First-Order Equations a. Review i. So far we’ve learned about two differential equations: 1. Separable ′ a. ???? = ???? ???? ???? ???? ) 2. Linear ′ a. ???? = ???? ???? ???? + ????(????) b. Section 2.6: Exact Differential Equations ???????? i. We can consider differential equations to be written as ???? ????,???? + ???? ????,???? ) = 0; ???????? where P and Q are functions of x (independent) and y (dependent) ii. Our goal is to allow solutions to be defined implicitly by equations of the form ???? ????,???? = ???? ???????? ???????? iii. We can let F(x,y) be a function, define the differential of F: ???????? =???????????????? + ???????????????? iv. Example 1: So just to solve for dF ???? ????,???? = ???? + ???? 2 ???????? ???????? ???????? = ???????? + ???????? ???????? ???????? ???????? = 2???????????? + 2???????????? 1. Now based off of what we know, consider ???? ????,???? = ???? + ???? and ???? ????,???? = ( ) 2 2 ???? → ???? + ???? = ????; where C is positive. These can correspond to the level curves of F ( ) v. The level sets defined by ???? ????,???? = ???? are called integral curves of the differential equations vi. Example 2: ???? = 3???? + ???? ???????? + √ ???????????? 1. If there is a function F(x,y) such that ???????? = ???? ????,???? ???????? + ???? ????,???? ???????? = ????, then w is an exact differential form vii. If F is differentiable then: ???? ???? ???? ???? = ???????????????? ???????????????? ???? = ???? ???????? ???????? viii. Example 3: ???? ????,???? = ???? + ???? 2 ???????? ???? ???? = 2???? → = 0 ???????? ???????????????? ???????? ???? ???? = 2???? → = 0 ???????? ???????????????? ix. Exact Differential Equations: The equation ???? = ???? ????,???? ???????? + ???? ????,???? ???????? = 0 is an ???????? ???????? exact differential equation if w is exact. That being said, then = ???????? ???????? x. Example 4: Verify that ???? + ???? = ???? gives a solution to ???? = ???????????? + ???????????? = 0. Is the equation exact? ???? ????,???? = ???????? ????,???? = ???? **We need the partial of P and the partial of Q to be equal to each other** ???????? ???????? = 0 = 0 ???????? ???????? ???????? ???????? = ???????? ???????? 1. Is it using implicit differentiation? ???? + ???? = ???? ′ 2???? + 2???????? = 0 ???????? 2???? + 2???? = 0 ???????? ???????? ???? + ???????????? = 0 ???????????? + ???????????? = 0 2. This is an example of integral curves 2 2 xi. Integral Curves: The level set is defined by ???? + ???? = ????, so when solving for y, we are given integral curves that can contain multiple solutions curves… 2 2 ???? ???? = √???? − ???? ???? ???? = − ???? − ???? xii.Example 5: Given all we know now, solve 2???????????? + 4???? ???????? = 0 1. First, let’s decide what kind of differential equation it is? Check if it’s exact by identifying P and Q 3 3 2???????????? + 4???? ???????? = 0;????ℎ???????????????? = 2???????????????????? = 4???? ???????? ????2???? ???????? ???? 4???? 3) = = 0 = = 0 ???????? ???????? ???????? ???????? **Since they are equal to each other, we can verify they are exact** 2. Now we need to find some function F(x,y)=C such that… ???????? = ???? = 2???? ???????? ???????? 3 = ???? = 4???? ???????? ???? ????,???? = ∫2???????????? + ????(????) 3. We can go on to solve for the function of y (????(????)) ???????? ???????? = 4???? = ????′(????) ???????? ???????? ???????? …???? ???? = 4???? 3 ( ) 3 ???? ???? = ∫ 4???? ???????? 4 ???? ???? = ???? ???? ????,???? = ???? + ???? ???? → ???? ????,???? = ???? + ????2 4 4. Then we are left with our solution which is defined by ???? ????,???? = ???? ???? + ???? = ???? Suggested Homework: Section 2.6: 4, 9, 10, 12, 19, 23, 26, 28 Finish Worksheet 4

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