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# Last Week of Notes Math 340

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This 2 page Class Notes was uploaded by Susan Ossareh on Saturday May 7, 2016. The Class Notes belongs to Math 340 at Colorado State University taught by in Spring 2016. Since its upload, it has received 8 views. For similar materials see Intro-Ordinary Differen Equatn in Math at Colorado State University.

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Date Created: 05/07/16

Math 340 Final Lectures – Introduction to Ordinary Differential Equations – May 2 to 6 What We Covered This Week: 1. Congrats guys, we made it! This last week was pure review and we worked through 3 worksheets. Because they are considered homework, I can’t go over the problems however I’ll rewrite each worksheet below so you can work through them 2. Before doing that, we did take a quiz this week a. Covered sections 5.2, 5.3, 5.4 3. Worksheet 11 ???? ′ 2???? ′ a. Consider the initial value problem ???? − 4???? − 5???? = ???? ,???? 0 = −1,???? 0 = 0 ( ) i. Apply the Laplace Transform to both sides of the equation and express it as an equation on s and Y(s) ii. Use partial fraction decomposition to express Y(s) as a sum of simple fractions iii. Use the inverse Laplace Transform to fin the solution to the IVP ???? ′ b. Consider the initial value problem ???? + 4???? = 2cos 3???? ,???? 0 = 1,???? 0 = −1 ( ) i. Apply the Laplace Transform to both sides of the equation and express it as an equation on s and Y(s) ii. Use partial fraction decomposition to express Y(s) as a sum of simple fractions iii. Use the Inverse Laplace Transform to find the solution to the IVP c. When you factor the term Y(s) on step 1, can you see a pattern for the polynomial that multiplies Y(s)? 4. Worksheet 12 a. Consider the Initial value problem ???? − ???? − 2???? = ???? 2????with y(0)=0 and y’(0)=0 i. Solve the initial value problem using the method of undetermined coefficients ii. Solve the problem using the Laplace Transform iii. The solutions found in part 1 and 2 should be equal, is there any other solutions to the initial value problem? Why? iv. Write the IVP as a linear system of equations in matrix form 1. Find the fundamental matrix for the associated homogeneous system 2. Classify the equilibrium points of this system 3. Use the variation of parameters method to find a particular solution to the initial value problem. (Notice that the initial condition is ???? 0 =( ) ???? (0,0) ) 5. Worksheet 13 a. In Worksheet 12 we found the fundamental matrix for ???? = ( ′ 0 1)????. Use Y(t) to compute 2 1 e where A is the coefficient matrix of this system ′′ ′ 3 2 b. Consider the second order nonlinear differential equation ???? + ???? ???? − 1 − ???? − ???? = 0 ) i. Convert this equation into a first order system ii. Find all nullclines and equilibrium points of the system −1 1 c. Consider the linear system x’=Ax where ???? = ( −2 1) i. Find the real valued solution to the equation ii. Classify the equilibrium point as stable, asymptotically stable, unstable and as a saddle, center, nodal or spiral iii. Identify the corresponding phase plane and draw the arrows indicating the orientation of the curve d. True or False? ′ ???????????? (????) i. The constant function ???? ???? = ???? is a solution of ???? ???? = ( ) 1+???? 2 ii. The equation ???????? − 1 ???????? + ???? − ???????? ???????? = 0 is exact 2 iii. If ???? − ???? 0 ???? = 0 holds true, then (???? − ???? ????) ????0= 0 is true also iv. Three given vector functions ???? 1 = 0,0,???? ,???? ???? =2cos ???? ,−sin ???? ,0 ,???? ???? = ] 3( ) [sin ???? ,cos ???? ,0] are linearly dependent on the interval (-π, π) −2 0 0 v. For the given matrix ???? = [ 1 −2 0 ], the algebraic multiplicity of the 0 1 −2 eigenvalue ???? = −2 is three vi. For the given matrix (same as above) the geometric multiplicity of the eigenvalue ???? = −2 is three vii. Consider the ODE system ???? ???? = ???? − 2???? ???????????? ???? ???? = 2???? − ????. The equilibrium point at the origin is a center Suggested Homework: Study for the final o Sections from Exam 1 o Sections from Exam 2 o Sections 9.9, 4.5, 5.1, 5.2, 5.3, 5.4 o Practice final from spring 2015

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