Practice Exam 1 Circle

NAME: Exam 1 Answer Sheet for Problems 1 through 5 1. Circle One: A B C D 3. Circle all that apply: A B C D 4. Circle One: A B C D 5. Circle all that apply: A B C D 2. y(0.5) = 1 2 1. Which of the ODEs has the following direction field? (a) (b) (c) (d) dy dx dy dx dy dx dy dx = −xy = xy = −x − y =x+y 3 2. Use Euler’s Method with n = 2 to approximate y(0.5) for the following IVP. dy = xy, y(0) = 2 dx Note that you should round your FINAL ANSWER ONLY to 3 decimal places. Any intermediate rounding that you do should be to at least 8 decimal places to ensure you get the same thing that I do. 4 3. Select all of the following that correctly describe the ODE below. Note that some algebra may be required. 2yx − y dy = dx 3 + x − x2 (a) This is separable. (b) This is first-order linear. (c) This is exact. (d) This is not an ODE. 4. Find and classify the equilibrium solutions of the following autonomous equation. dy = (y + 3)3 (y − 3)2 dx (a) y = −3 is stable, y = 3 is unstable (b) y = −3 is semistable, y = 3 is stable (c) y = −3 is unstable, y = 3 is semistable (d) None of the above. 5 5. Consider the IVP below. dy √ = xy, y(x0 ) = y0 dx Select all of the following for which the Existence and Uniqueness Theorem guarantees us a unique solution. (a) (x0 , y0 ) = (1, 1) (b) (x0 , y0 ) = (−1, 1) (c) (x0 , y0 ) = (0, 0) (d) (x0 , y0 ) = (−1, −1) 6 6. Solve the following IVP as a separable equation. 2y − 3 dy = , y(0) = 2 dx 1 − 2x Express your answer as an explicit solution. Follow the steps below. (a) Get all of your y terms on the LHS of your equation using Algebra. (+1 point) (b) Integrate properly to find an implicit solution. (+6 points) 7 (c) Use Algebra to solve for y and find an explicit solution. (+1 point) (d) Plug in your initial value to find the value of your constant. (+1 point) (e) Write your final answer with the value of the constant substituted in. (+1 point) 8 7. Solve the following IVP as a first-order linear equation. 2y − 3 dy = , y(0) = 2 dx 1 − 2x Express your answer as an explicit solution. Follow the steps below. (a) Put the ODE in first-order linear form. (+1 point) (b) Identify p(x) and q(x). (+2 points) (c) Compute µ(x) using the formula that I gave you. (+2 points) 9 (d) Compute y(x) using the formula that I gave you. (+4 point) (e) Plug in your initial value to find the value of your constant and then write your final answer with the value of the constant substituted in. (+1 point) 10 8. Solve the following IVP as an exact equation. 2y − 3 dy = , y(0) = 2 dx 1 − 2x Express your answer as an explicit solution. Follow the steps below. (a) Put the ODE in differential form (DF) and identify M (x, y) and N (x, y). (+1 points) (b) Use the Test for Exactness to show that this DF is exact. (+2 points) 11 (c) Use the definition of an exact equation to find F (x, y). (+4 point) (d) Use the fact that dF = 0 to find an implicit solution to the DF. (+1 point) 12 (e) Solve for y to find an explicit solution. (+1 point) (f) Plug in your initial value to find your constant C. Then substitute this back in for your final answer. (+1 point) 13