Let's break down and solve the problem step by step.

1. Writing the Input Function $f(t)$

The problem specifies that the load applied is $f(t) = 1$ for $1 \leq t < 10$ and zero elsewhere. This can be expressed using unit step functions $u(t)$ as:

$f(t) = u(t - 1) - u(t - 10)$

2. Laplace Transform of the Differential Equation

Given the differential equation:

$\frac{d^3 y(t)}{dt^3} + 3\frac{d^2 y(t)}{dt^2} + 10\frac{dy(t)}{dt} + 8y(t) = f(t)$

Taking the Laplace transform of both sides (assuming zero initial conditions), we have:

$s^3 Y(s) + 3s^2 Y(s) + 10s Y(s) + 8 Y(s) = \frac{1}{s} \left( e^{-s} - e^{-10s} \right)$

Simplifying the left-hand side:

$Y(s) \left( s^3 + 3s^2 + 10s + 8 \right) = \frac{1}{s} \left( e^{-s} - e^{-10s} \right)$

So,

$Y(s) = \frac{1}{s(s^3 + 3s^2 + 10s + 8)} \left( e^{-s} - e^{-10s} \right)$

3. Inverse Laplace Transform to Find $y(t)$

To find $y(t)$, we'll compute the inverse Laplace transform of $Y(s)$.

Since this involves partial fractions and using known transforms, this part is generally done with symbolic computation tools or by using tables of inverse Laplace transforms. Due to the complexity, let's focus on using MATLAB for the next steps.

4. MATLAB Code for Plotting $y(t)$

We can use MATLAB to compute the inverse Laplace transform and plot the solution.

matlabCopy code
syms s t Y
f_s = (exp(-s) - exp(-10*s)) / (s * (s^3 + 3*s^2 + 10*s + 8));
Y_s = ilaplace(f_s, s, t);
fplot(Y_s, [0, 35])
title('Solution y(t)')
xlabel('Time t')
ylabel('y(t)')
grid on

5. Re-write as a System of First-Order Differential Equations

Let's rewrite the third-order differential equation as a system of three first-order equations. Define:

$x_1 = y(t), \quad x_2 = \frac{dy(t)}{dt}, \quad x_3 = \frac{d^2 y(t)}{dt^2}$

Then the system becomes:

$\frac{dx_1}{dt} = x_2$ $\frac{dx_2}{dt} = x_3$ $\frac{dx_3}{dt} = f(t) - 3x_3 - 10x_2 - 8x_1$

6. MATLAB Code Using ode45

We'll use ode45 to solve the system of first-order differential equations.

matlabCopy code
function dydt = system_eqs(t, y)
    u = (t >= 1) - (t >= 10); % Define the input f(t)
    dydt = [y(2);
            y(3);
            u - 3*y(3) - 10*y(2) - 8*y(1)];
end

tspan = [0 35];
y0 = [0 0 0]; % Initial conditions [y(0), dy/dt(0), d^2y/dt^2(0)]
[t, y] = ode45(@system_eqs, tspan, y0);

% Plotting
figure
plot(t, y(:, 1))
title('Solution y(t) using ode45')
xlabel('Time t')
ylabel('y(t)')
grid on

This script will solve the system and plot the result.

---

Would you like a detailed explanation for any of these steps? Here are some related questions you might find useful:

1. How do you use Laplace transforms to solve differential equations?
2. What are the steps to perform a partial fraction decomposition?
3. How does the ode45 function in MATLAB work for solving ODEs?
4. Can you explain the process of converting a higher-order differential equation into a system of first-order equations?
5. What are some typical applications of the unit step function in engineering?

Tip: When using Laplace transforms, always ensure that you correctly handle the initial conditions, as they can significantly affect the solution.