Question

When using inverse Laplace uses the ilaplase module in Matlab. Below is an example for F(s) (5-5) F (S) = s(s+ 2)2 symsts F=(5-5)/(s*(s+2)^2); ilaplace(F) ans = -5/4+7/2*t+5/4)*exp(-2*t) simplify(ans) ans = 2 Page -5/4+7/2*t*exp(-2*t)+5/4*exp(-2*t) pretty(ans) - 5/4 + 7/2 t exp(-2 t) + 5/4 exp(-2 t) Which corresponds to f(t) ft() =-1.25+3.5te-2 +1.25e-2
Using example for MATLAB code in part 2

Answer 1

syms t
syms s
syms Fa(s)
syms Fb(s)
syms Fc(s)
syms Fd(s)
syms Fe(s)

syms fa(t)
syms fb(t)
syms fc(t)
syms fd(t)
syms fe(t)

% for function (a)
Fa(s)=s/(s*(s+1)*(s+5));

% for function (b)
Fb(s)=(3*s+2)/(s^2+2*s+10);

% for function (c)
Fc(s)=(3*s^2+6*s+6)/((s+1)*(s^2+10));

% for function (d)
Fd(s)=exp(-s)/(s^2);

% for function (e)
Fe(s)=((s^2+3*s+10)*(s+5))/((s+3)*(s+4)*(s^2+2*s+100));

% inverse of laplace function (a)
f(t)=ilaplace(Fa);
fa(t)=simplify(f);

% inverse of laplace function (b)
f(t)=ilaplace(Fb);
fb(t)=simplify(f);

% inverse of laplace function (c)
f(t)=ilaplace(Fc);
fc(t)=simplify(f);

% inverse of laplace function (d)
f(t)=ilaplace(Fd);
fd(t)=simplify(f);

% inverse of laplace function (e)
f(t)=ilaplace(Fe);
fe(t)=simplify(f);

% pretty(f) of function (a)

fprintf('Inverse laplace for function (a)\n');
pretty(fa);

% pretty(f) of function (b)

fprintf('Inverse laplace for function (b)\n');
pretty(fb);

% pretty(f) of function (c)

fprintf('Inverse laplace for function (c)\n');
pretty(fc);

% pretty(f) of function (d)

fprintf('Inverse laplace for function (d)\n');
pretty(fd);

% pretty(f) of function (e)

fprintf('Inverse laplace for function (e)\n');
pretty(fe);

% for graph create T and S

T=linspace(0,10,101);
S=linspace(1,25,101);

% for corresponding f(t) using T [ for function (a) ]
fa_t=zeros(1,length(T));

% for corresponding f(t) using T [ for function (b) ]
fb_t=zeros(1,length(T));

% for corresponding f(t) using T [ for function (c) ]
fc_t=zeros(1,length(T));

% for corresponding f(t) using T [ for function (d) ]
fd_t=zeros(1,length(T));

% for corresponding f(t) using T [ for function (e) ]
fe_t=zeros(1,length(T));

% for corresponding F(s) using S [ for function (a) ]
Fa_s=zeros(1,length(S));

% for corresponding F(s) using S [ for function (b) ]
Fb_s=zeros(1,length(S));

% for corresponding F(s) using S [ for function (c) ]
Fc_s=zeros(1,length(S));

% for corresponding F(s) using S [ for function (d) ]
Fd_s=zeros(1,length(S));

% for corresponding F(s) using S [ for function (e) ]
Fe_s=zeros(1,length(S));

% calculate all f(t)
for c=1:length(T)
fa_t(c)=eval(fa(T(c)));
fb_t(c)=eval(fb(T(c)));
fc_t(c)=eval(fc(T(c)));
fd_t(c)=eval(fd(T(c)));
fe_t(c)=eval(fe(T(c)));
end

% calculate all F(s)
for c=1:length(S)
Fa_s(c)=eval(Fa(S(c)));
Fb_s(c)=eval(Fb(S(c)));
Fc_s(c)=eval(Fc(S(c)));
Fd_s(c)=eval(Fd(S(c)));
Fe_s(c)=eval(Fe(S(c)));
end

% plot f(t) and F(s) for (a)
figure(1)
subplot(1,2,1);
plot(T,fa_t);
ylabel('f(t)');
xlabel('t');
title('f(t) vs t for (a)');

subplot(1,2,2);
plot(S,Fa_s);
ylabel('F(s)');
xlabel('s');
title('F(s) vs s for (a)');

% plot f(t) and F(s) for (b)
figure(2)
subplot(1,2,1);
plot(T,fb_t);
ylabel('f(t)');
xlabel('t');
title('f(t) vs t for (b)');

subplot(1,2,2);
plot(S,Fb_s);
ylabel('F(s)');
xlabel('s');
title('F(s) vs s for (b)');

% plot f(t) and F(s) for (c)
figure(3)
subplot(1,2,1);
plot(T,fc_t);
ylabel('f(t)');
xlabel('t');
title('f(t) vs t for (c)');

subplot(1,2,2);
plot(S,Fc_s);
ylabel('F(s)');
xlabel('s');
title('F(s) vs s for (c)');

% plot f(t) and F(s) for (d)
figure(4)
subplot(1,2,1);
plot(T,fd_t);
ylabel('f(t)');
xlabel('t');
title('f(t) vs t for (d)');

subplot(1,2,2);
plot(S,Fd_s);
ylabel('F(s)');
xlabel('s');
title('F(s) vs s for (d)');

% plot f(t) and F(s) for (e)
figure(5)
subplot(1,2,1);
plot(T,fe_t);
ylabel('f(t)');
xlabel('t');
title('f(t) vs t for (e)');

subplot(1,2,2);
plot(S,Fe_s);
ylabel('F(s)');
xlabel('s');
title('F(s) vs s for (e)');

f(t) vs t for (b) F(s) vs s for (b) 0.45 0.4 0.35 0.3 F(S) 0.25 0.2 0.15 0 2 4 6 8 10 O 5 10 15 20 25

f(t) vs t for (c) F(s) vs s for (c) f(t) F(s) R 4 6 8 10 0 5 10 15 20 25

f(t) vs t for (d) F(s) vs s for (d) f(t) F(s) 0.05 0 2 4 6 8 10 0 5 10 15 20 25 S

f(t) vs t for (e) F(s) vs s for (e) 0.055 0.05 0.045 f(t) 0.04 0.035 -0.6 0.03 -0.84 0 2 4 6 8 10 0 5 10 15 20 25

f(t) vs t for (a) F(s) vs s for (a) 0.14 0.09 0.12 0.07 0.06 0.05 F(s) 0.04 0.02 0.01 0 2 4 6 8 10 do 5 10 15 20 25 S