Laplace Transform

Question 1

Laplace Transform

Answer 1

L[f(t)] = ƒ(s) = ∞ ∫ 0 f(t)e^(-st) dt

Question 2

parameter differentiation

Answer 2

L[t] = ∞ ∫ - ∞ te^(-st) dt
= ∂/∂s ∞ ∫ 0 te^(-st) dt
= -∂/∂s 1/s = 1/s^2

Question 3

partial fractions method

i.e.

s+3/(s+1)(s+2)(s+3)

Answer 3

take top of fraction for LHS and some constants for finding the coefficients

s+3 = a/(s+1) + b/(s+2) + c/(s+3)

multiply out

s+3 = a(s+2)(c+3) + b(s+1)(s+3) + c(s+1)(s+2)

solve for roots i.e. s = -2 , s = -1 and s = -3

Question 4

Inverse Laplace Transform

Answer 4

find f(t) using partial fractions

use table to work backwards

Question 5

Laplace transform of a derivative

to find the nth derivative

Answer 5

L[dy/dt] use integration by parts

second derivative can be expressed in terms of first

L[f’] = -f(0) + sL[f]

= -f’(0)+sf(0) + s^2L[f]

for nth derivative

L[f^(n)] =s^n L[f] - n Σ i=1 s^(n-i) f^(i-1) (0)

Question 6

nth derivative

Answer 6

L[f^(n)] =s^n L[f] - n Σ i=1 s^(n-i) f^(i-1) (0)

Question 7

Solution of ODE’s using Laplace Transform

Answer 7

Laplace Transform of both sides.

evaluate @ boundary conditions

(often use partial fractions)

use the table

Question 8

f(t) = (t ∫ 0) d/dt f(u)du

Answer 8

= g’(t)

Question 9

applications of laplace transform

Answer 9

in electric circuits to simplify differential equations

Question 10

to take a laplace transform of sin or cos

Answer 10

turn into exponentials

i.e. e^(it) - e^(-it)

then take the laplace transform of each seperately

L[e^(it)] - L[e^(-it)]