midterm2 - Name SECOND PRACTICE TEST FOR MIDTERM 1 MATH 216...

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Name: SECOND PRACTICE TEST FOR MIDTERM 1, MATH 216 Problem 1. Answer the following questions. a) 5 pts. Consider ( tanx ) y 0 + 1 x + 2 y = x 3 Fine the largest interval of x containing x = - 1 where a unique solution of this initial-value problem is guaranteed to exist based on what we have learned in the course. Explain your answer. Do not solve the diﬀerential equation. b) 5 pts Consider the autonomous system dx/dt = 7 x (5 - x ) Draw a phase diagram and determine whether x = 0 is a stable or unstable equilibrium. c) 5 pts Consider the initial value problem dy/dx = - 1 / (1 + x 2 ) , y (0) = 1 Suppose we apply the improved Euler method to approximate the solution on the interval [0 , 2] with step size h = 0 . 02. Comparing the result of this numerical calculation with the exact solution at x = 2 suppose that the cumulative error at x = 2 is 0.2. From this information, predict a step size which might be reasonably expected to result in a cumulative error 1/4 that obtained for h = 0 . 02 . d) 5 pts Consider

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midterm2 - Name SECOND PRACTICE TEST FOR MIDTERM 1 MATH 216...

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