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

This preview shows pages 1–5. Sign up to view the full content.

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 , .

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

View Full Document
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 , ;
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.

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: 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: ....
View Full Document

{[ snackBarMessage ]}

### Page1 / 5

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

This preview shows document pages 1 - 5. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online