Use the fact that L {earín, F(s-a) to show that L{e-at sin kt} = (Hk4F-and
12.21 a) Given that F(s)f(t)), show that d F(s) b) Show that d"F(s) ds" c) Use the result of (b) to find Ptt5), L(t sin Bt), and Pte cosh t
(2pt) 2. Based on the fact that sin(x) = 0 when x = kT for all integers k, Euler wrote sin(x) as an infinite product as follows: sin(r)(1 1 + 1 + 37 1+ 27T _ _ 37T TT T 2 1- 4T2 9T2 167T2 Follow Euler's example and write cos(x) as an infinite product of the type above, so that cos(a) (1 2 (1 - c2a2)(1 - c3a2)(1 - c4a2)... (2pt) 2. Based on the fact that sin(x) =...
use Theorem 7.2 to find L{f(t)} (i have pictured the table of 7.2) ** just solve #26 & #30 please!! NOT 28** thank you!! 26. f(t) = (2t - 1) 28. f(t) = t - e-9 + 5 30. f(t) = (e' - e-)2 THEOREM 7.2 Transforms of Some Basic Functions (a) L{1} = 1 (b) L{t"} = 1 n = 1, 2, 3,... (C) L{e} = 1 (e) L{cos kt} = 2 * 2 (() {{sinh k} = 1...
3. 8p] Show that the force field F(x,y, z) sin y, x cos y + cos z, -y sin z) is conservative and use this fact to evaluate the work done by F in moving a particle with unit mass along the curve C with parametrization r(t (sin t, t, 2t), 0 <t<T/2. 4. 8p] A thin wire has the shape of a helix x = sin t, 0 < t < 27r. If the t, y = cos t,...
5. Use the fact that T(s)[(1 – ) = / sin is to prove that 2T |T(1/2+it)| = V whenever t ER. ett te-nt"
Problem 5. Let F(r,y) (e-v-v sinzy) ?-(ze-s + z sin zyj (1) Show that F is a gradient field. (2) Find a potential function f for it (3) Use the potential function f to evaluate F-ds, where x is the path x(t) = (t,t2) for 0sts1. (NO credit for any other method.)
Differential equation Q5 please / Q10 if possible, thank you so much Exercises 1. Show that L{cos kt) = s22 for s> 0. 2. Euler's formula elkcos kt+i sin kt can be used to obtain an additional formula cos kt(ek+e-ik), Show that the result of Exercise 1 can now be obtained with a formal application of the Laplace transform. 3. Obtain the transform for sin kt by an argument similar to the one suggested in Exercise 2 4. Evaluate L(r2...
Use the table provided and the fact that L ( w ″ ) = s 2 W ( s ) − s w ( 0 ) − w ′ ( 0 ) and L ( w ′ ) = s W ( s ) − w ( 0 ) (where W ( s ) is the Laplace transform of w) to solve the initial value problem w ″ + w = t 2 + 2 where w ( 0 )...
Elementary Laplace Transtorms Y(S) = {f} -L e-stf(t)dt fc = C-'{F(s)} F(s) = {f} f(t) =-'{F(s)) F(s) = {f} -CS 1. 1 1 12. uct) le S> 0 S> 0 . s S 2. eat 1 13. ucOf(t-c) e-csF(s) S> a S-a n! 3. t",n e Z 14. ectf(t) F( sc) S> 0 sh+1 4. tP, p>-1 (p+1) S> 0 SP+1 15. f(ct) F). c>0 16. SFt - 1)g(t)dt F(s)G(*) 5. sin at S> 0 16. cos at 17. 8(t...
Differential equations 7.3 Operational properties I Table for reference if needed. Use operational properties of the Laplace Transform to show Hint: F(t)=1.5(1) t S+1 t TABLE OF LAPLACE TRANSFORMS f(0) L{f(0) = F(s) f(t) L {f(0)} = F(s) 1. 1 20. eat sinh kt k (s – a) - R2 S 1 s- a 2. t 21. ear cosh kt 52 (s - a)- K 3. " n! +10 n a positive integer 22. tsin kt 2ks (52 + 2)2...