7 L 'LJ' :! OR? T * [4 ! 715 n rip ti up qi x r' '=(91nmnvmic :I;J = D Spring 2017 Stony Brook University Snliyaisachi Muklusrjeo, Matthew Dnnnonlmrg Problem Set 6 Differential Equations with Linear Algebra Homework Problems 61. Computing Inverse Laplace Transforms. (7+7+7+9 points) (a) Find the inverse Laplacc transforms of the following functions (you may use any stan- dard formula involving Laplace transforms listed in the book): an (Cl (b) Find "((;g+l-0;F) 6.2 Initial Willie Problems via anluce Tl'misform. (10+1[) points) (a) Solve the following initial value problem using Laplace transform: xv" + 4x' + 13x = te"; x(0) = 0, x'(0) = 2. (b) Use the convolution theorem to derive the indicated solution x(t) of the given initial value problem: sc" + 4.1:' + [3:1: = f(t); 2:(0) = 0, x'(0) = 0. m(t) = g/u [(t - u)e'2" sin(3'u)du. 6.3 Laplace Tl'mwform of Discoutimmus Filtration. (10 points) (a) Define the Heaviside function 0 fort<0 HI. = U {1 forOSt. Show that [H(t - a.)](s) = fie'". (b) Solve the differential equation y" = HU - a) (0 < a), with initial conditions y(0) = 1, y'(0) = 0- 6.4. (Bonus Problem) Bump Function. (it) points) Prove that the function Mm) = exp for Iml < l 0 otherwise is everywhere differentiable and that its derivative is continuous. (In faint, the function has continuous derivatives of all orders. This is an example of a bump function.) Due Date: Thursdny, March 23, at the beginning of recitation.f ( t ) = L -1 {F ( s )} 1. 1 3. t n , n = 1, 2,3,K 5. 1 s n! s n +1 p t 3 2 7. sin ( at ) 9. t sin ( at ) 11. Table of Laplace Transforms F ( s ) = L { f ( t )} sin ( at ) - at cos ( at ) 2s a 2 s + a2 2as (s 2 + a2 ) 2a 3 2 2 2 cos ( at ) - at sin ( at ) 15. sin ( at + b ) 17. sinh ( at ) 19. e at sin ( bt ) 21. e at sinh ( bt ) 23. t ne at , n = 1, 2,3,K 25. uc ( t ) = u ( t - c ) 2 2 (s - a) 2 + b2 b (s - a) 2 -b (s - a) n +1 27. uc ( t ) f ( t - c ) e - cs s - cs e F (s) 29. ect f ( t ) F ( s - c) 31. 1 f (t ) t t f ( t - t ) g (t ) dt 33. 35. f (t ) 37. f ( n) ( t ) 0 2 n! s 4. t p , p > -1 6. t 8. cos ( at ) 10. t cos ( at ) n - 12 , n = 1, 2,3,K 1 s-a G ( p + 1) s p +1 1 3 5L ( 2n - 1) p n+ 1 2n s 2 s 2 s + a2 s2 - a2 (s + a2 ) 2 sin ( at ) + at cos ( at ) F ( u ) du 14. cos ( at ) + at sin ( at ) 16. cos ( at + b ) 18. cosh ( at ) 20. e at cos ( bt ) 22. e at cosh ( bt ) 24. f ( ct ) 26. d (t - c ) Dirac Delta Function 28. uc ( t ) g ( t ) 30. t n f ( t ) , n = 1, 2,3,K 32. f ( v ) dv (s + a ) s ( s + 3a ) (s + a ) 2 2 2 f (t + T ) = f (t ) sF ( s ) - f ( 0 ) 36. f ( t ) 2 2 s cos ( b ) - a sin ( b ) s2 + a2 s 2 s - a2 s-a (s - a) 2 + b2 s-a (s - a) 2 - b2 1 s F c c e - cs e - cs L { g ( t + c )} ( -1) n F ( n) ( s ) F (s) s t 34. 2 2 0 F (s)G (s) 2 2as 2 2 s sin ( b ) + a cos ( b ) s2 + a2 a 2 s - a2 b e at F ( s ) = L { f ( t )} 2 2 Heaviside Function 2. 12. (s + a ) s(s - a ) (s + a ) 2 13. 2 f ( t ) = L -1 {F ( s )} T 0 e - st f ( t ) dt 1 - e - sT s 2 F ( s ) - sf ( 0 ) - f ( 0 ) s n F ( s ) - s n-1 f ( 0 ) - s n- 2 f ( 0 )L - sf ( n- 2) ( 0 ) - f ( n-1) ( 0 ) Table Notes 1. This list is not a complete listing of Laplace transforms and only contains some of the more commonly used Laplace transforms and formulas. 2. Recall the definition of hyperbolic functions. et + e - t et - e - t cosh ( t ) = sinh ( t ) = 2 2 3. Be careful when using \"normal\" trig function vs. hyperbolic functions. The only difference in the formulas is the \"+ a2\" for the \"normal\" trig functions becomes a \"- a2\" for the hyperbolic functions! 4. Formula #4 uses the Gamma function which is defined as G ( t ) = e - x x t -1 dx 0 If n is a positive integer then, G ( n + 1) = n ! The Gamma function is an extension of the normal factorial function. Here are a couple of quick facts for the Gamma function G ( p + 1) = pG ( p ) p ( p + 1)( p + 2 )L ( p + n - 1) = 1 G = p 2 G ( p + n) G ( p) u\fThis is how you get the partial fraction to question 1 f ( t ) = L -1 {F ( s )} 1. 1 3. t n , n = 1, 2,3,K 5. 1 s n! s n +1 p t 3 2 7. sin ( at ) 9. t sin ( at ) 11. Table of Laplace Transforms F ( s ) = L { f ( t )} sin ( at ) - at cos ( at ) 2s a 2 s + a2 2as (s 2 + a2 ) 2a 3 2 2 2 cos ( at ) - at sin ( at ) 15. sin ( at + b ) 17. sinh ( at ) 19. e at sin ( bt ) 21. e at sinh ( bt ) 23. t ne at , n = 1, 2,3,K 25. uc ( t ) = u ( t - c ) 2 2 (s - a) 2 + b2 b (s - a) 2 -b (s - a) n +1 27. uc ( t ) f ( t - c ) e - cs s - cs e F (s) 29. ect f ( t ) F ( s - c) 31. 1 f (t ) t t f ( t - t ) g (t ) dt 33. 35. f (t ) 37. f ( n) ( t ) 0 2 n! s 4. t p , p > -1 6. t 8. cos ( at ) 10. t cos ( at ) n - 12 , n = 1, 2,3,K 1 s-a G ( p + 1) s p +1 1 3 5L ( 2n - 1) p n+ 1 2n s 2 s 2 s + a2 s2 - a2 (s + a2 ) 2 sin ( at ) + at cos ( at ) F ( u ) du 14. cos ( at ) + at sin ( at ) 16. cos ( at + b ) 18. cosh ( at ) 20. e at cos ( bt ) 22. e at cosh ( bt ) 24. f ( ct ) 26. d (t - c ) Dirac Delta Function 28. uc ( t ) g ( t ) 30. t n f ( t ) , n = 1, 2,3,K 32. f ( v ) dv (s + a ) s ( s + 3a ) (s + a ) 2 2 2 f (t + T ) = f (t ) sF ( s ) - f ( 0 ) 36. f ( t ) 2 2 s cos ( b ) - a sin ( b ) s2 + a2 s 2 s - a2 s-a (s - a) 2 + b2 s-a (s - a) 2 - b2 1 s F c c e - cs e - cs L { g ( t + c )} ( -1) n F ( n) ( s ) F (s) s t 34. 2 2 0 F (s)G (s) 2 2as 2 2 s sin ( b ) + a cos ( b ) s2 + a2 a 2 s - a2 b e at F ( s ) = L { f ( t )} 2 2 Heaviside Function 2. 12. (s + a ) s(s - a ) (s + a ) 2 13. 2 f ( t ) = L -1 {F ( s )} T 0 e - st f ( t ) dt 1 - e - sT s 2 F ( s ) - sf ( 0 ) - f ( 0 ) s n F ( s ) - s n-1 f ( 0 ) - s n- 2 f ( 0 )L - sf ( n- 2) ( 0 ) - f ( n-1) ( 0 ) Table Notes 1. This list is not a complete listing of Laplace transforms and only contains some of the more commonly used Laplace transforms and formulas. 2. Recall the definition of hyperbolic functions. et + e - t et - e - t cosh ( t ) = sinh ( t ) = 2 2 3. Be careful when using \"normal\" trig function vs. hyperbolic functions. The only difference in the formulas is the \"+ a2\" for the \"normal\" trig functions becomes a \"- a2\" for the hyperbolic functions! 4. Formula #4 uses the Gamma function which is defined as G ( t ) = e - x x t -1 dx 0 If n is a positive integer then, G ( n + 1) = n ! The Gamma function is an extension of the normal factorial function. Here are a couple of quick facts for the Gamma function G ( p + 1) = pG ( p ) p ( p + 1)( p + 2 )L ( p + n - 1) = 1 G = p 2 G ( p + n) G ( p) u\fThis is how you get the partial fraction to question 1