# 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

This preview shows pages 1–4. Sign up to view the full content.

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 .

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

## 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.

### Page1 / 4

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

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online