0910F-108-02-midterm1-questions

0910F-108-02-midterm1-questions - y 1 = x 2 and y 2 = x 2...

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Duke University Fall 2009 MATH 108-02 ODE and PDE Midterm #1 : October 1, 2009. No calculator allowed. No books or notes allowed. You are allowed to use the back of the sheets. You have 75 minutes to complete this exam. Document your work. Clarity will be considered in the grading of this exam. 1. Solve y 0 + y 2 sin( x ) = 0. 2. Find the general solution to the equation y 00 - 8 y 0 + 12 y = 0. 3. Knowing that y 1 = x - 1 is a solution to x 2 y 00 - 2 y = 0, use the method of reduction of order to find a second solution to this equation. 4. Knowing that
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Unformatted text preview: y 1 = x 2 and y 2 = x 2 (ln x ) are solutions to x 2 y 00-3 xy +4 y = 0, nd the general solution to x 2 y 00-3 xy + 4 y = x 2 . (You may assume that we work in x > 0.) 5. Using an integration factor, solve the equation ( y + xy ) dx + xdy = 0. 6. Use the Laplace transform to solve the initial value problem y 00 + y = ( 1 , t < / 2; , t / 2; y (0) = 0 ,y (0) = 1 . 7. Prove that L{ f ( t ) } = s L{ f ( t ) } -f (0). 1...
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