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M334Sp06exam3-solutions

# M334Sp06exam3-solutions - Math 334(Ordinary Differential...

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Unformatted text preview: Math 334 (Ordinary Differential Equations) Exam 3 KEY Elementary Laplace Transforms f ( t ) = L- 1 { F ( s ) } F ( s ) = L{ f ( t ) } 1. 1 1 s , s > 2. e at 1 s- a , s > a 3. t n , n = positive integer n ! s n +1 , s > 4. sin at a s 2 + a 2 , s > 5. cos at s s 2 + a 2 , s > 6. sinh at a s 2- a 2 , s > | a | 7. cosh at s s 2- a 2 , s > | a | 8. e at sin bt b ( s- a ) 2 + b 2 , s > a 9. e at cos bt s- a ( s- a ) 2 + b 2 , s > a 10. t n e at , n = positive integer n ! ( s- a ) n +1 , s > a 11. u c ( t ) e- cs s , s > 12. u c ( t ) f ( t- c ) e- cs F ( s ) 13. e ct f ( t ) F ( s- c ) 14. Z t f ( t- τ ) g ( τ ) dτ F ( s ) G ( s ) 15. δ ( t- c ) e- cs 16. f ( n ) ( t ) s n F ( s )- s n- 1 f (0)- ··· - f ( n- 1) (0) 17. (- t ) n f ( t ) F ( n ) ( s ) Part I: Multiple Choice Mark the correct answer on the bubble sheet provided. 1. What is the radius of convergence of the power series ∑ ∞ n =1 n n ( x +1) n n ! ? a) 1 /e b) ( e- 1) /e c) 1 d) ( e + 1) /e e) e f) ∞ g) h) 3 Solution: a) 2. Find all of the singularities of ( x 4- x 3 ) y 00 + ( x 3- x 2 ) y + ( x 2 + x ) y = 0 , and determine whether each singularity is regular or irregular. a) x = 0 is an irregular singularity b) x = 0 is a regular singularity c) x = 0 is an irregular singularity, and x = 1 is a regular singularity d) x = 0 is a regular singularity, and x = 1 is an irregular singularity e) x = 0 and x = 1 are irregular singularities f) x = 0 and x = 1 are regular singularities g) x =- 1 ,x = 0, and x = 1 are irregular singularities h) x =- 1 ,x = 0, and x = 1 are regular singularities Solution: f) 3. What is a fundamental set of solutions of3....
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M334Sp06exam3-solutions - Math 334(Ordinary Differential...

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