Do not write in this space. Problem Possible Score Number Points Total 48

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1 MTH 337. Name MTH 337. Differential Equations Exam II March 15, 2019 T. Judson Do not write in this space. Problem Possible Score Number Points Total 48 Directions Please Read Carefully! You have 50 minutes to take this exam. Be sure to use correct mathematical notation. Any answer in decimal form must be accurate to three decimal places, unless otherwise specified. To receive full credit on a problem, you will need to justify your answers carefully unsubstantiated answers will receive little or no credit (except if the directions for that question specifically say no justification is necessary, such as in a matching or a / section). Please be sure to write neatly illegible answers will receive little or no credit. If more space is needed, use the back of the previous page to continue your work. Be sure to make a note of this on the problem page so that the grader will know where to find your answers. You are permitted one 4 6 inch note card (both sides). Calculators are allowed. Good Luck!!!

2 1. Indicate whether each of the following statements are true or false. No justification is [8] needed. (a) The equilibrium solutions of a first-order system occur at the intersection of the x and y-nullclines. (b) The origin is the only equilibrium solution for any 2 2 linear system. (c) If A is 2 2 matrix, then no solution of dx/dt = Ax can blow up (approach infinity) in finite time. (d) If v is an eigenvector for a 2 2 matrix A, then any nonzero multiple of v is also an eigenvector for A. (e) The function x(t) = (e 2t, e πt ) is never a solution to any 2 2 linear system dx/dt = Ax. (f) If A is 2 2 matrix, then the system dx/dt = Ax can can have three distinct straight-line solutions. (g) Euler s method is the optimal numerical algorithm for solving systems of differential equations. (h) Euler is pronounced as oiler. Solution: (a), (b), (c), (d), (e), (f), (g) (h). Page 2

3 2. Consider two tanks of water with each tank initially holding 100 gallons. Ten pounds [10] of salt has been dissolved in each tank. A brine solution containing 0.5 lbs of salt per gallon flows into the first tank at a rate of 3 gal/min and the brine mixture in this tank flows into the second tank at a rate of 5 gal/min. The brine mixture in the second tank flows back into the first tank at a rate of 2 gal/min and is simultaneously pumped out of the second tank at a rate of 3 gal/min. Set an initial value problem that describes this system. You do not need to solve this system. Solution: x = x y y = 1 20 x 1 20 y x(0) = 10 y(0) = 10. Page 3

4 3. (a) Find the general solution of the linear system [5] x = 2x + y y = 8x 5y Solution: If we write this system in matrix form, we have ( ) ( ) dx dt = x x(0) = The characteristic equation of A is det(a λi) = (2 λ)( 5 λ) 8 = λ 2 + 3λ 18 = (λ + 6)(λ 3) = 0. Therefore, the eigenvalues of A are λ 1 = 6 and λ 2 = 3. Furthermore, v 1 = (1, 8) is an eigenvector for λ 1 and v 2 = (1, 1) is an eigenvector for λ 2. Thus, the general solution to the system is ( ) ( ) 1 1 x(t) = αe 6t + βe 3t. 8 1 (b) Solve the initial value problem [5] x = 2x + y y = 8x 5y x(0) = 0 y(0) = 8 Solution: From (a), the general solution to the system is ( ) ( ) 1 1 x(t) = αe 6t + βe 3t. 8 1 To find α and β, we solve the system α + β = 0 8α + β = 8 to find that α = 8/9 and β = 8/9. Thus, the solution to our initial value problem is x(t) = 8 ( ) 1 9 e 6t + 8 ( ) e3t. 1 Page 4

5 (c) Sketch the phase portrait of the [5] x = 2x + y y = 8x 5y in the xy-plane. Sketch the straight-line solutions and sketch several solution curves including the solution curve through x(0) = 0 y(0) = 8 Label this solution curve and indicate the direction of the solution on this trajectory. y x Solution: Page 5

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7 4. Consider the system dx = x(2 x y) dt dy = y(y x). dt (a) Find the x and y-nullclines of the system. Sketch and label the nullclines on the [5] graph below. Be sure to indicate the direction of the solution on the nullclines. (b) Find all of the equilibrium solutions for the system. Label these on the graph below. [5] (c) Sketch the trajectory of a solution curve starting at x(0) = 3 and y(0) = 1 on the [5] graph below. y 4 2 x Solution: The x-nullclines (in red) are x = 0 y = x + 2. The y-nullclines (in green) are y = 0 y = x. The equilibrium solutions are (0, 0), (1, 1), and (2, 0). Page 7

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Do not write in this space. Problem Possible Score Number Points Total 100

Math 9. Name Mathematical Modeling Exam I Fall 004 T. Judson Do not write in this space. Problem Possible Score Number Points 5 6 3 8 4 0 5 0 6 0 7 8 9 8 Total 00 Directions Please Read Carefully! You