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Exam_1-1 - dy dx = 2 x 3 y y(0 = 10 To help in keeping your...

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Math 3323 - Exam 1 Name____________________ September 29, 2000 Put boxes around your answers to make them easier to find. 1. (20 points) Solve each of these first order linear initial value problems. (a) dy dx + 8 y = 34 e 9 x , y (0) = 8. (b) 1 100 dy dt + 1 t y = 1, y (1) = 0.
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2. (12 points) Consider the differential equation dy dx = y ! 5 x ! 2 . For which one(s) of the following initial conditions does the Fundamental Existence and Uniqueness Theorem guaranteee a unique solution? Explain why the theorem doesn’t apply at the other(s). (a) y(3) = 0. (b) y(0) = 5 (c) y(2) = 3
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3. (15 points) Solve the exact initial value problem 15 x 4 ydx + (2 y + 3 x 5 ) dy = 0, y (1) = 2. ( Note: It is actually exact--you don’t need to verify it. Just solve it.)
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4. (18 points) Use Euler’s method with step size h = 0.2 to estimate at x = 1 the value of the solution to the initial value problem
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Unformatted text preview: dy dx = 2 x + 3 y , y (0) = 10. To help in keeping your work organized, fill in the following chart with the values at the intermediate steps. x y 0.0 10. 0.2 0.4 0.6 0.8 1.0 5. (15 points) Find the general solution of the following differential equation, and write y as an explicit function of x. dy dx = y 4 5 6. (20 points) The separable differentiable equation dx dt + x 2 = x occurs in certain biological problems. (a) Find the general solution of this differentiable equation, with x written as an explicit function of t. For each of the following initial conditions, determine what happens to the solution of the initial value problem as t ! + " : Does it converge to !" , 0, 1, or + ! ? Explain your answer. ( Note: These parts of the problem can be answered without knowing the solution of the d.e. from part (a).) (b) x (0) = 0.1 (c) x (0) = ! 0.1...
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Exam_1-1 - dy dx = 2 x 3 y y(0 = 10 To help in keeping your...

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