
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) =...
Consider the Fermi–Dirac function, f(E) = 1∕[e(E−EF)∕kT + 1] . Define x = (E − EF)∕kT and hence show that f ′(x) = df (x)∕dx = −ex∕(ex + 1)2. (a) Plot f (x) versus x and y = ∣ f ′(x)∕f ′(0)∣ vs. x. (b) What are f and y at x = ±2? What does the interval Δx = 4 about x = 0 represent? (c) Show that the width Δx of the y vs. x curve between the...
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
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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...