Question

Find 1d. (LT = Laplace Transforms) use the function:

>>pretty(laplace(exp(-a*t)*cos(b*t)))

For la – d, find the LTs of the functions by hand, and then use MATLAB to verify the LTs. Include your hand calculations and your MATLAB script for the report. la) 2e-3t + 4te-5t + 6t2 b) 7e-atu(t – 2) c) t(sin(wt)) d) t-e-at

Answer 1

for the following functions; we need to calculate the Laplace tramform of the following ternition by and then we need to use HATLAB to verify the results Da) x (t) = 2 e 3t+ At 65t + 6t ; we need to calculate X (5) : x(s): «[X(t)] = [283446+5+ 6+2) - d[281] + [ate st] XLSX the Laplace ham form of a[234] - šta the laplace hamlom of a[test] = 6753 & the laplace tram form of a [ +²] = 2 23. hence; x(s) = 3.+ + 5)2 + the to lowing matlab Command can be used to verify the above result.

clear all
clc

syms t
%% equation 1a
x1 = 2*exp(-3*t)+4*t*exp(-5*t)+6*t^2;
%% find laplace transform
laplace(x1)

Db) X(t) = Je-at u (t-2): we need to calculate x(s) : (5)=d[x(4)] > «[teu Ct -») = 4 d xe 25«fe-a (4*)] = 7e 25 x 2 29 2 [e-at] = 7e 2 (Sta) + ista) the following matlab Command can be used to verify the above result

%% equation 1b
syms t a
x2 = 7*exp(-a*t)*heaviside(t-2);
%% find laplace transform
laplace(x2)

DO) X(t) = t (Sin wt) Need to calculate X (S); *(s): 2 [2CH)] = + (Sinwe)] - the laplace transform of & (Sint) && [+] = 1/32; thus; & (S)- & [tsin (wt)] = - 2ws the following matlab Command can be used to verify the above result. (5²+62) 2

%% equation 1c
syms t w
x3 = t*sin(w*t);
%% find laplace transform
laplace(x3)

Dd) X(t)= t²e at Need to calculate X (S); X(5)=d[ą (t)] - « [t*e*at] d [4) - * *[e-at]. Štahenee, [t'e-ot] the following matlab Code Can be used to verify the above result.

%% equation 1c
syms t a
x4 = t^2*exp(-a*t);
%% find laplace transform
laplace(x4)