A Laplace transform calculator converts a function of time, f(t), into a function of s, F(s), and back again with the inverse transform. Laplace transforms turn differential equations into algebra problems, which makes them essential in engineering, physics and control systems.
The definition is F(s) = ∫₀^∞ e^(−st) f(t) dt. Common results include: the transform of 1 is 1/s, of t is 1/s², of tⁿ is n!/sⁿ⁺¹, of e^(at) is 1/(s − a), of sin(bt) is b/(s² + b²) and of cos(bt) is s/(s² + b²). These table entries handle most homework problems.
Solving a differential equation: transform both sides, using the rule that the transform of y′ is sY(s) − y(0) and of y″ is s²Y(s) − s y(0) − y′(0). Solve for Y(s), then use partial fractions and the table to find the inverse transform. For example, y′ + 2y = 0 with y(0) = 3 becomes sY − 3 + 2Y = 0, so Y = 3/(s + 2) and y(t) = 3e^(−2t).
Useful properties: linearity, the first shifting theorem (multiplying by e^(at) shifts s to s − a), and the second shifting theorem for step functions. Common mistakes include forgetting initial conditions and mistakes in partial fractions.
Tips for getting accurate results from the Laplace Calculator: double-check that every input uses the right units, enter numbers without commas or symbols, and run the calculation a second time with a quick estimate in your head to confirm the answer looks reasonable.
How to use the Laplace Calculator on Minedit: enter your values in the fields, press calculate, and review the step-by-step breakdown under the result. If something looks off, change one input at a time to see how it affects the answer, which is also a great way to learn the math behind it.
Learn more on Wikipedia’s Laplace transform article. Find Fourier transform, integral and Euler’s method calculators on Minedit.