Supplemental Instruction: Math 267. SI Leader Ron Drs. Castillo-Gil, Hentzel, Sacks August 30th, 2015 Chapters 1.1 and 1.2

Transcription

1 Supplemental Instruction: Math 267 SI Leader Ron Drs. Castillo-Gil, Hentzel, Sacks August 30th, 2015 Chapters 1.1 and 1.2

2 The Plan Introductions Ice Breakers Question of the Day Types, Orders, Linearity of Differential Equations Examples Initial Value Problems Examples Questions/Comments/Concerns? Exit Ticket/Feedback Next Time... in SI

3 A joke... if it s funny

4 Ice Breaker Time! Turn to someone near you and introduce yourself. Talk about: Your name Major Hometown What you like about math? Maybe? (Why you re taking the class)

5 Question of the Day QoD: What are differential equations? What do you think we can use them for? Differential equations are equations that contain derivatives with respect to one or more independent variables. We can use DEs for mathematical models in real life applications. What about the normal form of a differential equation? The normal form of a DE is: (dy/dx) = f(x,y), for a first-order DE. In general, it would be (d n y/dx n ) = f(x,y,y,... y n ), for an nth-order DE.

6 Types, Orders, Linearities What are the types of differential equations? Ordinary & Partial. Ordinary Differential Equations are known to have derivatives with respect to only one independent variable. Partial Differential Equations involve partial derivatives with respect to two or more independent variables. What does it mean to refer to an order of a differential equation? The order of a DE refers to the highest derivative in the equation. How can you determine the linearity of a differential equation? The dependent variable y and its derivatives y, y,... y n are of the first power and the coefficients a 0, a 1,... a n, depend only on the independent variable x.

7 Ex: Practice Questions So, let s practice: What order are these differential equations? 1) t 5 y (6) + t 3 y + 2y = 0 2) (dy/dx) 4 - xy + 3 = 0 Are these differential equations linear or non-linear? 3) (y - 1)y + x = e 3x 4) xy + 3(dy/dx) = 72

8 Implicit/Explicit Solutions What is an implicit solution? An implicit solution is a solution that is not just expressed in terms of the independent variable x. The dependent variable y and the x tend to be together to form a result (i.e. x 2 + y 2 = 25). What is an explicit solution? An explicit solution is a solution in which the dependent variable y is shown directly in terms of the independent variable x. It shows the direct relationship between x and y (i.e. y = (⅓)x + 7).

9 Ex: Practice Questions Are these differential equations implicit or explicit? 5) y = (1/16)x 4 6) x 2 + y 2 = 25 7) y = cx + cos(x 2 ) 8) x(dy/dx) + x 5 y = 75e 3x + 2

10 Initial Value Problems What are initial value problems? What can they normally include? Initial value problems are problems that give you initial conditions for a certain equation. It often comes with differential equations, a general family of solutions, and initial conditions set to help find particular solutions. What do they want you to solve for? (Hint: Something in particular?) Particular solutions that are specific to those initial conditions. Adding on, what is an interval of definition (a.k.a. the interval of existence, the interval of validity, and the domain of the solution)? The interval is where the dependent variable y can exist, in terms of the independent variable x (i.e. from one to infinity).

11 Ex: Practice Questions Some practice IVPs: Given the DE, y + 2xy 2 = 0, y = 1/(x 2 + c) is a one-parameter family of solutions. Using the given initial conditions, find the particular solution of the differential equation: 9) y(2) = ⅓ 10) y(1/2) = -4

12 Questions/Comments/Concerns Anything... anything at all.

13 Exit Ticket/Feedback On your index card, please give me some feedback for this first lesson: 1. Did I miss any of your questions? 2. Is there anything that you would have preferred me to go over in better detail? 3. What can I do better to make this easier to learn for you? 4. Any favorite parts? (If you have none, that s okay. </3)

14 Next Lesson Our next lesson will be on Monday, August 31st. Location: Carver 0184 Time: 6:10 pm- 7:00 pm We will be covering the real life application of differential equations, such as mixtures, finance, population dynamics, etc., in mathematical models. Whoo! See you all next time! :D