Question

1. Use Laplace transforms to solve the following differential equations for ?(?) for ? ≥ 0. Use ?(0) = 0 and ?̇(0) = 1 for each case.

i. 0 = ?̈(?) + 2?̇(?) + 4?(?)

ii. 0 = ?̈(?) + 3?̇(?) + 2?(?)

iii. 5 = ?̈(?) + 5?̇(?) + 6?(?)

3. For the three differential equations from problem one determine the steady-state value of the system using:

a. lim?→0 ??(?),

b. lim ?→∞ ?(?) analytically,

c. lim ?→∞ ?(?) using MATLAB

Please solve #3 and show MATLAB code

Answer 1

(i)

clear;
clc;
t = [0 10];
y0 = [0 1];
[t,y] = ode45(@myfun,t,y0);
plot(t,y(:,1))
xlabel('Time (s)')
ylabel('y(t)')
fprintf('The steady-state value is %.2f.\n',y(end,1))

function ydot = myfun(t,y)
ydot = [y(2);-2*y(2)-4*y(1)];
end

The steady-state value is -0.00.

(ii)

clear;
clc;
t = [0 10];
y0 = [0 1];
[t,y] = ode45(@myfun,t,y0);
plot(t,y(:,1))
xlabel('Time (s)')
ylabel('y(t)')
fprintf('The steady-state value is %.2f.\n',y(end,1))

function ydot = myfun(t,y)
ydot = [y(2);-3*y(2)-2*y(1)];
end

The steady-state value is 0.00.

(iii)

clear;
clc;
t = [0 10];
y0 = [0 1];
[t,y] = ode45(@myfun,t,y0);
plot(t,y(:,1))
xlabel('Time (s)')
ylabel('y(t)')
fprintf('The steady-state value is %.2f.\n',y(end,1))

function ydot = myfun(t,y)
ydot = [y(2);5-5*y(2)-6*y(1)];
end

The steady-state value is 0.83.