test1_spring_05 - Name Answers MA-366 Spring 2005 Test 1...

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MA-366, Spring 2005, Test 1 Name: Answers Note: You may use scratch paper. One sheet (both sides) of notes are allowed. No calculators. Problem 1. (12 points) Solve the initial value problem dy dx D x 3 NUL 2y x ; y . NUL 1 / D 0 : and determine the largest interval where the solution is defined. Answer: This is a linear equation. The general solution is y . x / D C x 2 C x 3 5 . The solution of the IVP is y . x / D 1 5x 2 C x 3 5 , defined for x < 0 . 1
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MA-366, Spring 2005, Test 1 2 Problem 2. (10 points) Find the general solution to the equation . 1 C e x / DC2 dy dx C y DC3 D 1 Answer: This is a linear equation again. The solution is y . x / D Ce NUL x C e NUL x ln . 1 C e x / .
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MA-366, Spring 2005, Test 1 3 Problem 3. (12 points) Solve the initial value problem and find the largest interval where the solution is defined p 1 C x 2 dy dx D xy 3 ; y . 0 / D NUL 1 : Answer: Separable equation. Solution: y D NUL DLE 3 NUL 2 p 1 C x 2 DC1 NUL 1 = 2 , j x j < p 5 = 2 .
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MA-366, Spring 2005, Test 1 4 Problem 4. (12 points) Consider the equation x 0 D f . x / , where f . x / is sketched below Find the equilibrium solutions, classify them by stability and sketch the phase portrait of this equation.
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