Laplace Transform Calculator
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The Laplace transform turns a function of time f(t) into a function of s: \(F(s) = \int_{0}^{\infty} e^{-st} f(t)\,dt\). Because it is linear, you transform each term with a table rule. For example, \(\mathcal{L}\{t^{2} + e^{3t}\sin 2t\} = \frac{2}{s^{3}} + \frac{2}{(s-3)^{2} + 4}\).
\(f(t) = t^{2} + e^{3 t} \sin\left(2 t\right)\)Use t. Supports t^n, sqrt(t), e^(at), sin(bt), cos(bt), sinh, cosh and products like t e^(2t) or sin(2t)cos(3t).
| Combined into one fraction | \(F(s) = \frac{2s^{3} + 2s^{2} – 12s + 26}{s^{5} – 6s^{4} + 13s^{3}}\) |
| Converges for | \(s > 3\) |
| Numeric check | \(\int_{0}^{\infty} e^{-st} f(t)\,dt\) at two values of s matches F(s) ✓ |
Show steps
- Write f(t) as a sum of basic terms: \(f(t) = t^{2} + e^{3t} \sin\left(2t\right)\)
- The Laplace transform is linear, so transform each term and add: \(\mathcal{L}\{a f + b g\} = a F(s) + b G(s)\).
- \(\mathcal{L}\left\{t^{2}\right\} = \frac{2}{s^{3}}\) using \(\mathcal{L}\{t^{n}\} = \frac{n!}{s^{n+1}}\)
- \(\mathcal{L}\left\{e^{3t} \sin\left(2t\right)\right\} = \frac{2}{\left(s – 3\right)^{2} + 4}\) using \(\mathcal{L}\{\sin(bt)\} = \frac{b}{s^{2}+b^{2}}\text{ with } s \to s – 3\)
How to use the Laplace transform calculator
- Type f(t) using t, for example t^2 + e^(3t) sin(2t).
- Press Transform. The calculator splits f into basic terms, applies a table rule to each, and adds the results.
- It also combines the answer into a single fraction, gives the values of s where the transform exists, and checks F(s) against the defining integral numerically.
Definition
\(\mathcal{L}\{f(t)\} = F(s) = \int_{0}^{\infty} e^{-st} f(t)\,dt\)
The integral exists for s larger than the growth rate of f. The transform is linear: \(\mathcal{L}\{af + bg\} = aF(s) + bG(s)\).
Table of common Laplace transforms
| f(t) | F(s) | Valid for |
|---|---|---|
| \(1\) | \(\frac{1}{s}\) | s > 0 |
| \(t^{n}\) | \(\frac{n!}{s^{n+1}}\) | s > 0 |
| \(e^{at}\) | \(\frac{1}{s-a}\) | s > a |
| \(\sin(bt)\) | \(\frac{b}{s^{2}+b^{2}}\) | s > 0 |
| \(\cos(bt)\) | \(\frac{s}{s^{2}+b^{2}}\) | s > 0 |
| \(\sinh(bt)\) | \(\frac{b}{s^{2}-b^{2}}\) | s > |b| |
| \(\cosh(bt)\) | \(\frac{s}{s^{2}-b^{2}}\) | s > |b| |
| \(t\sin(bt)\) | \(\frac{2bs}{(s^{2}+b^{2})^{2}}\) | s > 0 |
| \(t\cos(bt)\) | \(\frac{s^{2}-b^{2}}{(s^{2}+b^{2})^{2}}\) | s > 0 |
| \(\sqrt{t}\) | \(\frac{\sqrt{\pi}}{2s^{3/2}}\) | s > 0 |
The shifting rule
Multiplying by \(e^{at}\) shifts the transform: \(\mathcal{L}\{e^{at}f(t)\} = F(s-a)\). So \(\mathcal{L}\{e^{2t}\cos 3t\} = \frac{s-2}{(s-2)^{2}+9}\) and \(\mathcal{L}\{t^{3}e^{-2t}\} = \frac{6}{(s+2)^{4}}\).
Products of sines and cosines are first rewritten as sums, for example \(\sin^{2}t = \frac{1}{2} – \frac{1}{2}\cos 2t\).
Why Laplace transforms are useful
Derivatives become multiplication by s: \(\mathcal{L}\{y^{\prime}\} = sY(s) – y(0)\) and \(\mathcal{L}\{y^{\prime\prime}\} = s^{2}Y(s) – sy(0) – y^{\prime}(0)\). A linear differential equation becomes an algebra problem in s, which you solve for Y(s) and then invert. See the inverse Laplace and differential equation calculators.
Laplace transform practice quiz: 10 questions
Test yourself with 10 multiple-choice questions. Choose an answer and press Submit to see if you are right and why. Your score appears at the end.
Quiz complete
Frequently asked questions
What is the Laplace transform?
It converts f(t) into F(s) = ∫ from 0 to ∞ of e^(−st) f(t) dt. It turns derivatives into multiplication by s, which makes linear differential equations much easier to solve.
Which functions can this calculator transform?
Sums and products of powers of t (including √t), exponentials e^(at), sin, cos, sinh and cosh with arguments like 3t. Products of sines and cosines are converted to sums automatically.
How is the answer checked?
The calculator evaluates the defining integral numerically at two values of s and compares it with the formula result.
Why is there a condition like s > 3?
The integral only converges when e^(−st) decays faster than f(t) grows. For e^(3t), that needs s > 3.
How do I go back from F(s) to f(t)?
Use the inverse Laplace transform calculator, which splits F(s) into partial fractions and reads each piece from the table.
Sources
- Paul’s Online Math Notes: The Definition of the Laplace Transform
- Paul’s Online Math Notes: Table of Laplace Transforms
- NIST Digital Library of Mathematical Functions §1.14: Integral Transforms
- MIT OpenCourseWare 18.03: Differential Equations
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- Differential Equation Calculator
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