The Laplace transform
The Laplace transform turns differential equations into algebra. It is the engineer's favourite tool for circuits, control systems, and switched inputs.
Definition
The Laplace transform of \(f(t)\) is \(F(s) = \mathcal{L}\{f\} = \int_0^\infty e^{-st}f(t)\,dt\), defined for values of \(s\) where the integral converges. It converts a function of time \(t\) into a function of a new variable \(s\). For \(f = 1\), \(\mathcal{L}\{1\} = \frac{1}{s}\) for \(s > 0\).
\mathcal{L}\{f\}(s) = \int_0^\infty e^{-st} f(t)\,dtA table of basic transforms
\(\mathcal{L}\{e^{at}\} = \frac{1}{s - a}\); \(\mathcal{L}\{t^n\} = \frac{n!}{s^{n+1}}\); \(\mathcal{L}\{\sin bt\} = \frac{b}{s^2 + b^2}\); \(\mathcal{L}\{\cos bt\} = \frac{s}{s^2 + b^2}\). Most problems are solved by combining these entries with a few rules, rather than by integrating directly.
Linearity
\(\mathcal{L}\{af + bg\} = aF + bG\). So \(\mathcal{L}\{3e^{2t} - 4\sin t\} = \frac{3}{s - 2} - \frac{4}{s^2 + 1}\). Linearity is what makes transforms practical.
Transforms of derivatives
Integration by parts gives \(\mathcal{L}\{f'\} = sF(s) - f(0)\) and \(\mathcal{L}\{f''\} = s^2F(s) - sf(0) - f'(0)\). Differentiation in \(t\) becomes multiplication by \(s\), and the initial conditions are built in automatically. This is the key to solving initial value problems.
\mathcal{L}\{y''\} = s^2Y - sy(0) - y'(0)The shifting theorem
Multiplying by an exponential shifts the transform: \(\mathcal{L}\{e^{at}f(t)\} = F(s - a)\). So \(\mathcal{L}\{e^{-2t}\cos 3t\} = \frac{s + 2}{(s + 2)^2 + 9}\). This handles damped oscillations directly.
Existence
The transform exists for functions that are piecewise continuous and of exponential order, meaning \(|f(t)| \le Me^{ct}\) for large \(t\). Almost every function in applications qualifies, though \(e^{t^2}\) grows too fast to have a Laplace transform.
Worked example: computing a transform
Find \(\mathcal{L}\{te^{3t}\}\). From the table, \(\mathcal{L}\{t\} = \frac{1}{s^2}\). Shifting by \(a = 3\) gives \(\frac{1}{(s - 3)^2}\). Rules plus a table replace a page of integration by parts.
Common mistakes
Forgetting initial conditions in the derivative formulas. Mixing up the sine and cosine entries (sine has \(b\) on top; cosine has \(s\)). Shifting in the wrong direction: \(e^{at}\) gives \(s - a\), not \(s + a\).
Key takeaways
The Laplace transform \(\int_0^\infty e^{-st}f(t)\,dt\) converts time functions to \(s\)-functions. Use a table and rules: linearity, derivative formulas that include initial conditions, and the shift \(e^{at}f \to F(s - a)\). Derivatives become multiplication by \(s\), turning ODEs into algebra.
Flashcard
Laplace transform
Check your understanding
What is the Laplace transform of e^(5t)?