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The Laplace transform turns a function of time \(f(t)\) into a function of \(s\): \(F(s) = \int_0^\infty e^{-st} f(t)\,dt\). For example, \(\mathcal{L}\{\sin 2t\} = \frac{2}{s^2 + 4}\). It changes differential equations into algebra problems.
How to Use the Laplace Transform Calculator
- Type \(f(t)\), such as
t^2 e^(3t)or5t^3 - 2cos(4t). - Press Solve.
- Each step names the rule used: linearity, a table entry, the first shifting theorem, or multiplication by \(t^n\). The answer is checked against the integral computed numerically.
Table of Common Laplace Transforms
| \(f(t)\) | \(F(s)\) |
|---|---|
| \(1\) | \(\frac{1}{s}\) |
| \(t^n\) | \(\frac{n!}{s^{n+1}}\) |
| \(e^{at}\) | \(\frac{1}{s – a}\) |
| \(\sin(bt)\) | \(\frac{b}{s^2 + b^2}\) |
| \(\cos(bt)\) | \(\frac{s}{s^2 + b^2}\) |
| \(\sinh(bt)\) | \(\frac{b}{s^2 – b^2}\) |
| \(\cosh(bt)\) | \(\frac{s}{s^2 – b^2}\) |
| \(e^{at}f(t)\) | \(F(s – a)\) |
| \(t\,f(t)\) | \(-F'(s)\) |
Worked Examples
Example 1: \(\mathcal{L}\{5t^3 – 2\cos 4t\}\)
By linearity, transform each term. \(\mathcal{L}\{t^3\} = \frac{3!}{s^4} = \frac{6}{s^4}\), so \(5t^3 \to \frac{30}{s^4}\). \(\mathcal{L}\{\cos 4t\} = \frac{s}{s^2 + 16}\).
Answer: \(\frac{30}{s^4} – \frac{2s}{s^2 + 16}\).
Example 2: \(\mathcal{L}\{t^2 e^{3t}\}\)
\(\mathcal{L}\{t^2\} = \frac{2}{s^3}\). Multiplying by \(e^{3t}\) shifts \(s\) to \(s – 3\) (first shifting theorem).
Answer: \(\frac{2}{(s – 3)^3}\).
Example 3: \(\mathcal{L}\{e^{-t}\cos 2t\}\)
\(\mathcal{L}\{\cos 2t\} = \frac{s}{s^2 + 4}\). Shift \(s \to s + 1\): \(\frac{s + 1}{(s + 1)^2 + 4}\).
Example 4: \(\mathcal{L}\{t\sin t\}\)
\(\mathcal{L}\{\sin t\} = \frac{1}{s^2 + 1}\). Multiplying by \(t\) means taking \(-\frac{d}{ds}\): \(-\frac{d}{ds}\frac{1}{s^2+1} = \frac{2s}{(s^2 + 1)^2}\).
Why Engineers Use It
The transform of a derivative is algebraic: \(\mathcal{L}\{y’\} = sY(s) – y(0)\). So a differential equation like \(y’ + 2y = e^{-t}\) becomes an equation you can solve for \(Y(s)\). You then split \(Y(s)\) with partial fractions and use the table backwards.
Common Mistakes
- Shifting the wrong way. \(e^{at}\) replaces \(s\) with \(s – a\). For \(e^{-t}\), that is \(s + 1\).
- Forgetting \(n!\). \(\mathcal{L}\{t^3\} = \frac{6}{s^4}\), not \(\frac{1}{s^4}\).
- Transforming a product term by term. \(\mathcal{L}\{fg\} \ne \mathcal{L}\{f\}\mathcal{L}\{g\}\).
Learn the Math Behind the Laplace Transform Calculator
The Laplace transform turns a differential equation into algebra. It is defined by an improper integral from 0 to infinity.
Frequently Asked Questions
Does every function have a Laplace transform?
Most functions you meet in class do. The function must not grow faster than an exponential \(e^{ct}\). For example, \(e^{t^2}\) has no Laplace transform.
What does s mean?
\(s\) is a new variable, often thought of as a complex frequency. The transform exists for \(s\) larger than the function’s growth rate.
How is the answer checked?
The calculator computes \(\int_0^\infty e^{-st}f(t)\,dt\) numerically at two values of \(s\) and compares it with the formula. A match gives the ✓ Verified badge.
Related Guides
- Improper integral calculator
- Integration by parts (how the table entries are derived)
- Antiderivatives of exponential functions