Laplace Transform Differential Equations in MATLAB Programming

The Laplace Transform is a fundamental mathematical tool for analyzing dynamic systems in control engineering, electrical circuits, and mechanical vibrations. It converts time-domain differential equations into algebraic equations in the complex frequency domain ((s)-domain), making complex system responses significantly easier to solve.

In MATLAB, the Symbolic Math Toolbox allows you to perform Laplace transforms, solve for the (s)-domain transfer function, and apply the inverse Laplace transform to obtain exact time-domain analytical solutions using laplace(), ilaplace(), and solve().

Mathematical Foundation of the Laplace Transform

The Laplace transform converts a time-domain function (f(t)) into its (s)-domain equivalent (F(s)):

Using this transformation, time derivatives convert into algebraic terms involving initial conditions:

Step-by-Step Examples

Example 1: Solving a First-Order Linear Differential Equation

Consider the first-order differential equation:

syms y(t) s  % Define equation and symbolic Laplace representation Dy = diff(y, t);
eqn = Dy + 3*y == 6;
% Take Laplace transform of the equation Y = laplace(y, t, s);
eqnLaplace = laplace(eqn, t, s);
% Substitute symbolic Laplace term and initial condition y(0) = 2 eqnLaplace = subs(eqnLaplace, laplace(y, t, s), Y);
eqnLaplace = subs(eqnLaplace, y(0), 2);
% Solve for Y(s) in Laplace domain Y_s = solve(eqnLaplace, Y);
% Compute inverse Laplace transform to get y(t) y_t = ilaplace(Y_s, s, t)

Output:

y_t = 2 + 2*exp(-3*t)

This result shows an exponential decay settling toward a steady-state value of 2.

Example 2: Solving a Second-Order Differential Equation

Solve the second-order homogeneous differential equation:

syms y(t) s  % Define derivatives and differential equation Dy = diff(y, t);
D2y = diff(y, t, 2);
eqn = D2y + 5*Dy + 6*y == 0;
% Transform equation to s-domain Y = laplace(y, t, s);
eqnLap = laplace(eqn, t, s);
% Substitute initial conditions y(0) = 1 and y'(0) = 0 eqnLap = subs(eqnLap, laplace(y, t, s), Y);
eqnLap = subs(eqnLap, y(0), 1);
eqnLap = subs(eqnLap, subs(diff(y, t), t, 0), 0);
% Solve for Y(s) and compute inverse Laplace transform Y_s = solve(eqnLap, Y);
y_t = ilaplace(Y_s, s, t)

Output:

y_t = 3*exp(-2*t) - 2*exp(-3*t)

Plotting System Response in MATLAB

Visualize the resulting time-domain analytical solution over a 5-second window using fplot():

figure;
f
plot(y_t, [0, 5], 'LineWidth', 1.5, 'Color', 'b');
title('Response of Second-Order System y(t)');
xlabel('Time t (seconds)');
ylabel('Amplitude y(t)');
grid on;

Key Summary

• Use laplace() to convert time-domain differential equations into algebraic (s)-domain expressions.
• Use subs() to inject physical initial conditions (y(0)) and (y'(0)).
• Use solve() to isolate (Y(s)) algebraically.
• Use ilaplace() to convert the solved (s)-domain expression back to a time-domain equation (y(t)).
• Use fplot() to visualize transient and steady-state time responses.