Test1-1 - Answer using so we need and for both and to exist...

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MATH 315 FALL 2006 TEST 1 KEY 1. (15 pts.) Solve the differential equation , with . Answer: linear with integrating factor . Then , so , . Using , . Final solution: . 2. (15 pts.) Solve the differential equation , with . Answer: , so exact with . Then . Using , .
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Final solution: . 3. (15 pts.) Consider the differential equation , with . 1. Draw a rough sketch of the direction field for the equation, plotting direction field arrows for values (0,1/2), (0,1), (1/2,1/2), (1/2,1), (1,1/2), (1,1), (3/2,1/2), and (3/2,1). Write down the general formula used to calculate with the Euler Method. Answer: . Use the Euler formula to compute an approximate value for y(2), using a step size (you will need to take 2 steps). Answer: using first step: , , so , ;
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second step: , , , . 4. (7 pts.) Consider the differential equation . Determine the region in the - plane where there is a unique solution near each initial point in that region.
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Unformatted text preview: Answer: using , , so we need and for both and to exist. The -plane region is given by and or and . 5. (18 pts.) Consider the differential equation . 1. Determine and classify the equilibrium (critical) points for the equation. Answer: when and ; . and , so is unstable and is stable. 2. Determine the solution for the equation if y(0) = 2. Answer: , so separable . Now , so , and , or . Using , . Final solution: ; 6. (10 pts.) Solve the differential equation , with . Answer: , so linear with . Then , , and . Using , . Final solution: . 7. (15 pts.) Consider the differential equation . 1. Find the general solution for the equation. Answer: The characteristic equation is , so or ; general solution is . 2. Determine the Wronskian for fundamental solution set. Answer: . 3. Determine the solution that satisfies initial conditions , . Answer: and , so and , . Final solution: ....
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Test1-1 - Answer using so we need and for both and to exist...

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