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# Math 340 - Week 2 - Jan 25-29 Math 340

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This 9 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 33 views. For similar materials see Intro-Ordinary Differen Equatn in Math at Colorado State University.

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Date Created: 01/31/16

th Math 340 Lecture – Introduction to Ordinary Differential Equations – January 25 , 2016 What We Covered: 1. Worksheet 2: Direction Fields a. Highlights i. How direction fields correspond to distinct differential equations ii. You can get more practice with direction fields by visiting the Math 340 website 1. Under “Other Course Materials” you can download dfield 2. Course Content – Chapter 2: First-Order Equations a. Section 2.1: Differential Equations and Solutions Continued i. Direction Field: the geometric meaning of a differential equation and its solutions 1. Essentially, we use directional fields as a way to show the slope of the function at each point which then gives you a visual of the differential equation’s behavior ′ 2. Suppose you are given the diff eq: ???? = ????(????,????). We want to define this graphically which means we need to take into account the domain and range: ???? = { ????,???? |???? ≤ ???? ≤ ???? ???????????? ???? ≤ ???? ≤ ????}. Then there is the matter of defining the slopes of the equation at each point which is the solution curve: ???? ???? ) = ????(???? ,???? ) of which passes through the point (???? ,???? ). 0 0 0 0 0 3. Please refer to Figure 4 on page 21 of your text book for a specific example 4. **Note: horizontal functions are known as constant solutions** b. Section 2.2: Separable Functions i. The whole idea of separable functions is to group same variables on separate sides of the equal sign ???????? ii. Therefore the general form is???????? = ???? ???? ℎ(????) and then is solved so that y’s are on the LHS and t’s are on the RHS: ???????? = ???? ???? ???????? ℎ(????) ???????? ∫ = ∫???? ???? ???????? ℎ(????) iii. Example: Radioactive nuclei 1. The text book goes into all the science behind radioactivity and how to derive the differential equation, I’m skipping over those details because they aren’t necessarily relevant to separable equations. Instead we will just start off with the exponential equation ′ ???????? ???? = −???????? → = −???????? ???????? a. All we need to do is separate the variables ???????? = −???????? ???????? ???????? ∫ = ∫−???????????? ???? ln ???? = −???????? + ???? | | −????????+???? ???? = ???? |???? = ???? −???????????????? b. Now to solve for N we need to create a system of variables in order to account for the absolute value ???? ???? = {???? −???????????????? ???????? ???? > 0 {−???? −???????????????? ???????? ???? < 0 ???? i. Let ???? be interchangeable with A; where A isn’t equal to zero ???? ???? = ???????? −???????? c. This is the general solution that satisfies N(t) when it’s equal to and not equal to zero. Therefore, the can say the general solution is valid for any value of A. iv. The General Method: Let’s break down the steps of what we just covered in the example 1. Separate the variables 2. Integrate both sides 3. Solve for y (or, as seen in the radioactive example, N(t)) 4. Finally, check to see if ???? ???? = 0 is a solution Suggested Homework: Section 2.1: #2, 6, 8, 10, 12, 14 Section 2.2: #2, 6, 12, 16, 18, 24, 26 Study for quiz on Wednesday Jan 27 which will cover the sections 1.1-2.2 th Math 340 Lab – Introduction to Ordinary Differential Equations – January 26 , 2016 What We Covered: 1. Announcements a. Highlights i. Office Hours open this week 1. Office: Weber 15 2. Mon from 3-4 and Tues from 2-3 ii. Quiz tomorrow: covers sections 1.1 to 2.2 2. Course Content – Chapter 2: First-Order Equations a. Section 2.2: Solutions to Separable Equations Continued i. Example 1: Half-life 1. Half-life: the time take for the radioactivity of an isotope to fall to half of its original value 2. We will be using the formulas we worked with yesterday: ′ ???? = −???????? ???? ???? = ???????? −???????? 32 3. ???? is an isotope used in leukemia therapy. After 10hrs, 615mg of 1000mg is left over. Solve for the half-life of the substance… a. Based off of the information given, we can infer the initial condition is ???? 0 = 1000 → ???? = 1000 ???? ???? = 1000???? −???????? b. Then we can go about finding ???? ???? 10 = 615 −????(10) ???? 10 = 1000???? = 615 ln(0.615) ???? = = 0.04861 −10 So… −(0.04861)???? ???? ???? = 1000???? c. Now we have enough information to solve for time t ???????????????? − ???????????????? ???????????????????????? ????ℎ???????? ???? ???? = 500 −(0.04861)???? 500 = 1000???? 1 = ????− 0.04861 ???? 2 ∗∗ ???????????????????????????? ???????????? ???? ℎ???????????? − ????????????????,????ℎ???????? ???????????????????????????? ???????????? ????,???????????????? ???????????????????????????????? ????ℎ???????????????? ???????????????????????? ???????????????????? 1⁄ ∗∗ 2 ???? ≈ 14.3ℎ???????????????? ii. Example 2: Beer 1. A can of beer with the initial temperature of 40F is places in a room where the temperature is 70F. After 10 minutes, the temperature of the can increases to 50F. Find the temperature of the beer as a function of time. a. For this problem, we are going to need to take some physics into account. Newton’s Law of Cooling states that the rate of change of an object’s temperature (T) is proportional to the difference between its temperature and the ambient temperature (A). Therefore we are going to work with the equation: ???????? = −????(???? − ????) ???????? ???????? ????ℎ???????????? = ???????????????? ???????? ????ℎ???????????????? ???????? ???????????? ???? = ????ℎ???? ???????????????????????????????? ???????? ???????????????????????????????????????????????????????????? b. Our goal is to solve for T(t), to do this, we will separate the equation and then use the definite integral. Keep in mind that we also know T(0)=40 and t=0 ???????? = −???????????? (???? − ????) ???? ???? ???????? ∫ (???? − ????)= ∫ −???????????? 40 0 |???? − ????| ????????|???? − ????| = −???????? 0 *Because (T-A) and (T0-A) always have the same sign as each other, we can get rid of the absolute value bars** ???? − ???? ln ???? − ???? = −???????? 0 ???? −????????= ???? − ???? ???? − ???? −???????? 0 ???? (????0− ???? = ???? − ???? ???? = 40, ???? = 70 0 −???????? ???? ???? = ???? ????0− ???? + ???? ???? ???? = ???? −????????(−30 + 70 c. Solve for k 3 ???????? ???? = 2 = 0.405 10 d. Therefore the solution is… −0.405???? ???? ???? = −30???? + 70 b. Section 2.3: Models of Motion i. Linear Motion 1. We’ve already discussed one-dimensional motion in Chapter 1, so remembering back, there are a few kinematics equations that we can look at as differential equations ???? ???? a. ????????2 = −????,????:???????????????????????????????????????????????? ???????????????????????????????? 2 ???? ???? ???????? ???????? 2= ???????? = −???? ∫???????? = ∫−???????????? ???? ???? = −???????? + ???? ;????ℎ???????????? ???? = ???????????????????????????? ???????????????????????????????? ???????? 0 0 b. = −???????? + ???? 0 ???????? ∫???????? = ∫ −???????? + ???? ????0) 2 −???????? ???? ???? = 2 + ???? 0 + ????0;????ℎ???????????? ???? 0 ???????????????????????????? ???????????????????????????????? ii. Air Resistance 1. There’s obviously more forces acting on a free falling object than just gravity, like air resistance. There are some facts about air resistance that we need to take into account a. No motion = not resistance b. Motion = resistance always acts in the opposite direction 2. For our purposes, we will only consider the function of resistance to be dependent on velocity v ???? ???? = −????(????) × ???? 3. There are two models we need to keep in mind a. Air resistance is proportional to velocity (where r(v) is constant) ???? ???? = −???? × ???? ???? = ????????+ ???? ???? = −???????? − ???? × ???? b. Air resistance is proportional to ???? 4. Newton’s Second Law: The lesser known quote “May the force… be equal to mass times acceleration…” ???? = ???? × ???? ???????? ???? = ???? × ???????? ???????? −???????? − ???????? = ???? × ???????? ???????? ???????? = −???? − ???????? ???? *This is a separable equation so…** ???????? ???????? = ???????? −???? − ???? ???????? ∫ = ∫???????? −???? − ???????? ???? −???? ???????? ???? + ???? = × ln| − ???? − | ???? ???? ???????? −???????? ln|−???? − ???? | = ???? + ???? −???????? ???????? ???????? +???? = | − ???? − | ???? −???????? ???????? ???????? ???? = −???? − ???? Suggested Homework: Section 2.2: 2, 6, 12, 16, 18, 24, 26 Section 2.3: 2, 6, 8, 10, 13 Study for quiz th Math 340 Lecture – Introduction to Ordinary Differential Equations – January 27 , 2016 What We Covered: 1. Quiz 01 a. Highlights i. Administered in the last 15minutes of class ii.Covered content from 1.1-2.2 2. Course Content – Chapter 2: First-Order Equations a. Section 2.3: Models of Motion Continued i. Air Resistance 1. Based off of yesterday’s notes, we know the equation of force F is ???? = −???????? + ???? ???? = −???????? − ???????? 2. Use Newton’s second law in order to find the displacement function ???????? ???? = −???????? − ???????? ???????? ???????? ???????? = −???? − ???????? ???? 3. This is a separable equation so we can separate the variables to get: ???????? ????????= −???????? ???? + −???????? ???? ???? ???? ????−???? = | − ???? −????????| ???? −???????? ???? ???????? ???? = −???? − ???? ???? **A can be any value but we will rearrange the equation so it’s easier to −????????+???? work with. As of right now we have A???? . We can manipulate the −???????? variables and say that???? = ????** −???????? ???? ???? ???? = ???????? ???? − ???? ???? 4. From here, we can solve the equation using the limit to see if it approaches terminal velocity lim ????(????) ????→∞ −???????? ???? −???????? lim (???????? ???? − ???? ) = = ???????????? = ???????????????????????????????? ????????????????????????????????! ????→∞ ???? ???? ii. Example 1: Suppose you drop a brick from the top of a building that is 250m high. The brick has a mass of 2kg, and the resistance force is given by R=-4v. How long will it take the brick to reach the ground? What will be its velocity at that time? 1. For the first part of the problem, we will use the velocity equation: −???????? ???????? ???? = ???????? ???? − ???? 2. To solve for B, we can use the initial condition ???? 0 = 0 −????(0) ???????? ???? 0 = ???????? ???? − = 0 ???? ???????? (2)(9.8) ???? = = = 4.9 ???? 4 ???? ???? = 4.9???? −2????− 4.9 ???????? 3. We can now find the function x(t) such that ???? ???? = ???????? ???????? −2???? ????????= 4.9(???? − 1), it ∫???????? = ∫4.9 ???? ( −2????− 1 ???????? 4. Now we have our general solution…. −???? −2???? ???? ???? = 4.9( − ????) + ???? 2 5. We can find a particular solution using the initial condition ???? 0 = 250 −0 −???? ???? 0 = 4.9( 2 − 0) + ???? = 250 ???? = 252.45 −???? 2???? ???? ???? = 4.9( − ????) + 252.45 2 6. Therefore, at ???? ???? = 0 it reaches the ground at… −????2???? a. 4.9( 2 − ????) + 252.45 = 0 ???? = 51.5204 b. ???? 51.5204 = 4.9(???? −51.5204− 1) ≅ −4.9????/???? iii. Air Resistance Cases 1. Case 1: a. We know from before that R=-rv 2. Case 2: a. The magnitude of the air force is proportional to the square of the velocity 2 ???? ???? )|= |???????? | ???? ???? = −???? ???? ????;????ℎ???????????? − ???? ???? ???????? − ????(????) ???? ???? = −???? ???? ????) b. r(v) is either kv or –kv depending on the direction Suggest Homework: Section 2.3: #2, 6, 8, 10, 13 Math 340 Lecture – Introduction to Ordinary Differential Equations – January 29 , 2016 th What We Covered: 1. Quiz 02 a. Highlights i. So far covering Section 2.3 and 2.4 (which we will get to in today’s notes) ii. Be sure to do the suggested problems and utilize office hours 2. Course Content – Chapter 2: First Order Equations a. Section 2.3: Models of Motion Continued i. If you look back to Wednesday’s notes, we went over different cases of air resistance 1. Case 0: no air resistance 2. Case 1: ???? ???? = −???????? ???? ≥ 0 3. Case 2: magnitude of air resistance is proportional to the square of the velocity ii. Focusing on the second case, we can see the magnitude of R is | ( )| ( )| | 2 ???? ???? = ???? ???? ???? = ???????? 1. Using Newton’s Second Law, you can determine r(v) |−???? ???? ???? = ???? ???? ???? ;????ℎ???????????? ????(????) ≥ 0 2 ???????? = ???? ???? |????| ????|????| = ???? ???? |????| ???? ???? = ????(????) ????????:???? ???? = −????|????|???? 2. For ???? ≥ 0 ???? ???? = −???????? 2 3. Therefore… ???????? ???? = −???????? − ????|????|???? ???????? ???????? = −???? − ???? |????|???? ???????? ???? 4. We can then solve this using the separation of variables iii.Finding Displacement 1. To make these equations easier, we can get rid of the variable r by using the chain rule ???????? ???????? ???????? ???????? ???? = ???????? = ???????? ∗ ???????? = ???????? ∗ ???? ???????? ???? ???? = −???? − |????|???? ???????? ???? 2. Let’s apply this technique to an example… iv. Example 1: An ball of mass m=0.2kg is projected from the surface of the earth with velocity 0 = 50????/????. Assume that ???? = −????|????|???? for k=0.02. What is the max height reached by the object? 1. We know a few facts: ???? ≥ 0 → |???? = ????. Now we can go about trying to solve this problem using the “Finding Displacement” equation. 2 ???????????? = −???? − ???????? ???????? ???? 2. Then we separate the variables… ???????????? = − ???????? ???????? + ???????? 2 ???? 0 ???????????? ???????????? ???????? ∫ ???????? + ????????2 = − ∫ ???? 0 0 3. We can use u-sub to evaluate the integrals 2 ???????? 2???????? ???? ???? = ???? + ???? ???????? = ???? ???????? → ???????????? = 2???? 2 ???? ???? ???????? 2???? ln|????| =2???? ln|???? + ???? | 4. So then we get the general solution: ???? ????????2 ln(???? + ) = −???? + ???? 2???? ???? 5. Now we also know x=o and v=50, so we can plug these in to solve for x max ???? ????????0 ln(???? + ) = 0 + ???? 2???? ???? ???? ????2 ???? ????2 ???? = ln(???? + ???? 0) − ln(???? + ???? ) 2???? ???? 2???? ???? 2 ???? + ???? ????0 ???? ???? ???? = 2???? ln( ????2) ???? + ???? ???? ∗ ???? = max????ℎ???????? ???? = 0 ∗ ????????2 ???? ???? + 0 ???? = ln( ???? ) ???????????? 2???? ???? 2 ???? ????????0 ???????????????? = 2???? ln(1 +???????? ) ????????????????= 16.4???? b. Section 2.4: Linear Equations ′ i. The form of a first order linear equation is ???? = ???? ???? ???? + ????(????), which is a homogeneous equation where a(t) and f(t) are coefficients. 1. The equation is separable so we can solve for x ???????? = ???? ???? ???? ???????? ???????? ∫ =∫ ????(????)???????? ???????? ln ???? = ∫???? ???? ???????? |???? = ????∫???? ???? ???????????? ∫???? ???? ???????? ???? = ???????? ii.Example 1: Let’s solve for ???? = cos ???? ???? which is homogeneous and linear ???? ???? = ???????? ∫cos ???? ???????? sin(????) ???? ???? = ???????? 1. If ????(????) ≠ 0 then x’=a(t)x+f(t) is called inhomogeneous iii.Let’s practice finding the solution to inhomogeneous equations 1. Example 2: Newton’s law of cooling states that the rate at which an object losses or gains heat is proportional to the difference between the temperature of the object (T) and the temperature of the surrounding medium (A). That is illustrated in the equation: ???? = ???? ???? − ???? ;????ℎ???????????? ???? ???????? ????ℎ???? ???????????????????????????????? ???????? ???????????????????????????????????????????????????????????? a. We can rewrite the equation to fit the linear equation form ???? = ???????? − ???????? ′ ???? − ???????? = −????????;????ℎ???????? ???????? ???????????????????????????????????????????? ????ℎ???? ????ℎ???????????? ???????????????? *Multiply by an integrating factor which allows us to multiply ???????? the equation by a function u(t), like ???? , which will turn the left- hand side into the derivative of a product* −∫???????????? −????????+???? ???? = ???? ????−????????[???? − ???????? = −???? −???????????????? ????′????−????????− ???????? −???????????? = −???? −????????????????;????ℎ???????????? ???????? = ???? ,???? = ???? −????????,???????? −???????? = ???????? ,???????????? ???? = ???? *Then by the chain rule, we can set the equation up as…* ???????? −????????]′= −???? −???????????????? −???????? *Then to solve for T(t), let ???? = ???????? * ???? = ???? −???????????????? ( ) −???????? ???? ???? = ∫???? ???????? −???????? ???? ???? = ???????? ???????? −???????? = ???????? −???????? + ???? ???? = ???? [−???????? −???????? + ????] ???????? ???? ???? = −???? + ???????? iv. General Method 1. The general method of solving arbitrary linear equations using ???? = ???????? + ???? a. Rewrite the equation ′ ???? − ???????? = ???? b. Multiply the integrating factor so that the equation becomes: ???? ???? = ???? − ∫???? ???? ???????? ′ ′ (???????? ) = ???? ???? − ???????? = ???????? c. Integrate this equation to obtain: ???? ???? ???? ???? = ∫???? ???? ???? ???? ???????? + ???? d. Solve for x(t) Suggested Homework: Section 2.3: 2, 6, 8, 10, 13 Section 2.4: 2, 3, 4, 8, 11, 13, 14, 16, 36, 38

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