dynamical-systems-hw01-sols

dynamical-systems-hw01-sols - Instructor: Wayne Hacker Math...

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Unformatted text preview: Instructor: Wayne Hacker Math 550.391 Homework Set 1-f08 Name: Solutions Print your name neatly. If you forget to write your name, or write so sloppy that I can’t read it, you can lose up to 25 points! Answer all the questions that you can. Circle your answer where appropriate. • For problems 1-15 there is no partial credit. All problems worth 1 point each. • For problems 16-20 the problems are worth 2 points each. Note: You may only get 2 points if you have the correct answer! You must show your work. You will not receive credit for lucky guesses. Show your work as clearly as you can: if I can’t understand how you got an answer, I will not give you credit for it. DO NOT WRITE ON THIS PAGE IN THE SPACE BELOW 1 11 2 12 3 13 4 14 5 15 6 16 7 17 8 18 9 19 10 20 Math 550.391, Homework Set 1-f08, Hackernotes c Wayne Hacker 2008. All rights reserved. 2 Problems 1-4: Classification by type: Determine if the following equations are ordinary differential equations (ODE) or partial differential equations (PDE)? 1. u t = u xx ODE PDE (circle one) Answer: PDE 2. y 00 = y 2 y ODE PDE (circle one) Answer: ODE 3. u tt- u xx = 0 ODE PDE (circle one) Answer: PDE 4. y tt + xy xx = 0 ODE PDE (circle one) Answer: PDE Problems 5-7: Classification by order: Determine the order of the following equa- tions. 5. d 5 y dx 5 + dy dx 7- 7 y 9 = 0 order = 5 6. x 5 d 2 y dx 2 = y 3 order = 2 7. y = y 2 order = 1 Problem 8-10: Classification as linear or nonlinear: Indicate whether the given differential equations are linear or nonlinear. 8. y 00 = yy linear nonlinear (circle one) Answer: nonlinear 9. y 00 + | x | y = 0 linear nonlinear (circle one) Answer: linear 10. y = y 2 linear nonlinear (circle one) Answer: nonlinear Problem 11-13: Classification as autonomous or non-autonomous: Indicate whether the given differential equations are autonomous. 11. ˙ x = t autonomous non-autonomous (underline one) 12. ˙ x = sinh x autonomous non-autonomous (underline one) 13. y = y 2 autonomous non-autonomous (underline one) Math 550.391, Homework Set 1-f08, Hackernotes c Wayne Hacker 2008. All rights reserved. 3 Problem 14 and 15: Determine a solution: Indicate whether the given function is a solution to the differential equation. 14. Is y = e- x 2 a solution to the ODE y =- xy ? yes no (circle one) SOLUTION: Using the chain rule to differentiate y = e- x 2 we find y = e- x 2 (- 2 x ) =- 2 xy . Thus, e- x 2 is not a solution to the ODE. 15. For what values of ω is y = sin( ωx ) a solution to the ODE y 00 + 9 y = 0? SOLUTION: Using the chain rule to differentiate y = sin( ωx ) we find upon taking two derivatives y 00 =- ω 2 sin( ωx ) =- ω 2 y ⇒ y 00 + ω 2 y = 0 . For the ODEs to match we must have ω 2 = 9, or ω = ± 3....
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This note was uploaded on 02/28/2011 for the course MATH 550.391 taught by Professor Dr.hacker during the Fall '08 term at Johns Hopkins.

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dynamical-systems-hw01-sols - Instructor: Wayne Hacker Math...

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