function Lab3_prob3
clear; clc; close all;

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Problem 1
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%
% 1(c)
%%%%%%%%%%%%%%

% Defining a 1st order ODE
function dy=myODE(t,y)
    dy=-y+5*heaviside(t-2).*exp(-0.01*(t-1));
end

% Initial condition
y0=3;
% time interval
t=0:0.1:10;

% Solving the ODE 
[t,y1]=ode45(@myODE, t, y0);

%Defining a direction field of the ODE
[x,y]=meshgrid(t,-0.5:.1:5);
dy = myODE(x,y);
dx = ones(size(dy));
dyu = dy./sqrt(dx.^2+dy.^2);
dxu = dx./sqrt(dx.^2+dy.^2);

figure;
% Plotting the solution of the ODE
plot(t,y1, 'LineWidth', 3)
hold on;
% Plotting the direction field
quiver(x,y,dxu,dyu)
legend('solution of 1st order ode')


%%%%%%%%%%%%%%
% 1(e)
%%%%%%%%%%%%%%

%Finding the Laplace Transform of a function
t=sym('t');
f= heaviside(t-3)-heaviside(t-5)
laplace(f)

%Finding the Inverse Laplace Transform of a function
clear;
s=sym('s');
%%%%%% Modify this to answer 1(e)
F = 1/(s+5)+2*(exp(-2*s))/((s+5)*(s+1))
ilaplace(F)
figure;
ezplot(ilaplace(F),[0,10]), axis square, grid on
end