Test1-0 - a Show that the equation is not exact Answer Using you can see b Multiply the equation by and show that the new equation is exact Answer

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MATH 315 FALL 1998 TEST 1 SOLUTION KEY 1. (15 pts.) Solve the differential equation , with . Answer: This is linear so find the integrating factor . Then , so . Therefore . Using , we have , so and 2. (15 pts.) Solve the differential equation , with . Answer: The variables are already separated so use , where and . The solution is . We rewrite the equation as , and integrating gives us or . Using , we have , so , so the solution is given implicitly by 3. (15 pts.) Solve the differential equation .
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Answer: Rewriting the equation as , you can see that it is homogeneous. So let and then the equation becomes , or . Then , with and , so . Integrating gives us , so and the final solution is given implicitly by 4. (15 pts.) Solve the differential equation , with . Answer: The variables can be separated to give the differential equation or , where and , so . Then . Integrating gives us , or , so , and the final solution is given implicitly by 5. (15 pts.) Consider the differential equation
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Unformatted text preview: . a) Show that the equation is not exact. Answer: Using , you can see . b) Multiply the equation by and show that the new equation is exact. Answer: Using the new , you can see . c) Solve the equation for the solution that satisfies . Answer: You have , so , so and the final solution is given implicitly by 6. (10 pts.) Consider the differential equation . Determine the region in the plane where a unique solution should exist. Answer: Using , and , you need to determine where and/or do not exist. This happens if , so the region is determined by 7. (15 pts.) Consider the differential equation , with . a) Write down the general formula used to calculate with the Improved Euler Method. Answer: . b) Use the Improved Euler formula to compute an approximate solution to the equation at , using a step size (you will need to take two steps). Answer: If you use then the formula becomes . Starting with , , , and using , and using , . Therefore...
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This test prep was uploaded on 04/06/2008 for the course MATH 302 taught by Professor Goldberg during the Spring '08 term at Johns Hopkins.

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Test1-0 - a Show that the equation is not exact Answer Using you can see b Multiply the equation by and show that the new equation is exact Answer

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