Enter the differential equation to find its Laplace transformation.
Use this online Laplace remodel calculator to locate the Laplace transformation of a characteristic f(t). The calculator applies applicable Laplace transform system and imperative operations for the illustration.
Laplace transform is an absolute essential remodel that helps you to convert a function in actual variable (t) to a feature in complicated variable (s).
Mathematically,
\(F(s) = \int_{0}^{∞} f(t)e^{-st} dt\)
Where;
There are feasible strategies to discover Laplace transforms:
\(F(s) = \int_{0}^{∞} f(t)e^{-st} dt\)
Example:
Suppose we have the feature given underneath:
\(f(t) = 4e^{-2t} + e^{5t} + 3t^{2} - 7\)
Step 01: Write down the Laplace transform formula
\(F(s) = \int_{0}^{\infty} f(t)e^{-st} dt\)
Step 02: Input the given function \(f(t)\)
\(F(s) = \int_{0}^{\infty} (4e^{-2t} + e^{5t} + 3t^{2} - 7) e^{-st} dt\)
Step 03: Apply the formula to individual terms
\(\int_{0}^{\infty} 4e^{-2t} e^{-st} dt = 4\int_{0}^{\infty} e^{-(2+s)t} dt = 4\left[\frac{1}{-(2+s)}e^{-(2+s)t}\right]_{0}^{\infty} = \frac{4}{s+2}\)
\(\int_{0}^{\infty} e^{5t} e^{-st} dt = \int_{0}^{\infty} e^{(5-s)t} dt = \left[\frac{1}{5-s}e^{(5-s)t}\right]_{0}^{\infty} = \frac{1}{s-5}\) (valid for \(s > 5\))
\(\int_{0}^{\infty} 3t^{2} e^{-st} dt = 3\int_{0}^{\infty} t^{2}e^{-st} dt = 3\cdot \frac{2!}{s^{3}} = \frac{6}{s^{3}}\)
\(\int_{0}^{\infty} -7 e^{-st} dt = -7\int_{0}^{\infty} e^{-st} dt = -7\left[\frac{-1}{s}e^{-st}\right]_{0}^{\infty} = \frac{7}{s}\)
Step 04: Sum up all transforms together
\(F(s) = \frac{4}{s+2} + \frac{1}{s-5} + \frac{6}{s^{3}} + \frac{7}{s}\)
To transform returned this feature to the original time domain equation, you may use our inverse Laplace transform calculator.
| Property | Formula | Example Calculation |
|---|---|---|
| Laplace Transform | L{f(t)} = ∫₀⁺∞ e^(-st) f(t) dt | For f(t) = t, L{t} = ∫₀⁺∞ e^(-st) t dt = 1/s² |
| Laplace Transform of 1 | L{1} = 1/s | For f(t) = 1, L{1} = ∫₀⁺∞ e^(-st) dt = 1/s |
| Laplace Transform of e^(at) | L{e^(at)} = 1/(s - a) | For f(t) = e^(2t), L{e^(2t)} = 1/(s - 2) |
| Laplace Transform of t^n | L{t^n} = n!/s^(n+1) | For f(t) = t², L{t²} = 2!/s³ = 2/s³ |
| Laplace Transform of sin(at) | L{sin(at)} = a/(s² + a²) | For f(t) = sin(3t), L{sin(3t)} = 3/(s² + 9) |
| Laplace Transform of cos(at) | L{cos(at)} = s/(s² + a²) | For f(t) = cos(3t), L{cos(3t)} = s/(s² + 9) |
| Shifted Laplace Transform | L{e^(at) f(t)} = F(s - a) | For f(t) = t and a = 2, L{e^(2t) t} = F(s - 2) = 1/(s - 2)² |
| Laplace Transform of a Constant Multiple | L{cf(t)} = cL{f(t)} | For f(t) = 3t and c = 2, L{6t} = 6L{t} = 6/s² |
| Laplace Transform of Derivative | L{f'(t)} = sF(s) - f(0) | For f(t) = t, f'(t) = 1. L{f'(t)} = s(1/s²) - 0 = 1/s |
| Inverse Laplace Transform | L⁻¹{F(s)} = f(t) | If F(s) = 1/s², then L⁻¹{1/s²} = t |
Laplace transforms a time-variant signal into a frequency-related one, using complex math. It is used to simplify the process of solving differential equations.
“Laplace Transform Calculator enables one to submit a temporal function for its processing, resulting in its Laplace counterpart using typical transform equations.
Laplace substantially transforms aid in solving linear differential equations and scrutinizing mechanisms in disciplines such as engineering, physics, and control theory, by reducing the computing load.
Laplace transforms are widely used in electrical engineering, control theory, signal processing, and physics for examining dynamic systems and resolving ordinary differential equations.
reverse Laplace transform changes a function back to the time domain where the input is time, from the complex frequency domain where the input is the frequency.
You can put the function using normal math signs such as plus, minus, divide, and bracket pairs.
The calculator produces the Laplace transform of the specified function, denoted using complex variables (commonly 's' for the complex frequency).
The Laplace transformation of a fixed number 'c' is merely c/s, with 's' being the complex angular speed.
To transform an exponential function e^(at), where 'a' is a fixed number, use 1 divided by the difference between 's' and 'a'.
Certainly, the calculator can manage segmented functions by setting distinct equations for different time periods.
The impulse function is denoted by δ(t) in the temporal domain, with its Laplace transform equating the unity constant.
Absolutely, the Fourier transform of typical trigonometric functions, such as sine and cosine, can be determined. For example, the Laplace transform of sin(at) is a/(s^2 + a^2).
For a simple-shaped math equation like x^m, the Laplace change is m/s^(m+1), where m is a whole number greater than zero.
This is a calculator designed to incorporate initial conditions into the process of solving differential equations, which allows it to produce solutions that align with these starting points.
'The system's input-output correlation's Laplace transform is called the transfer function; this entity helps in scrutinizing the system's actions within the frequency realm. 'In the rewrite sentence, I have replaced various phrases with synonyms or closely related terms while keeping the message intact.
The Laplace remodel lets you convert a simple sign right into a complex frequency sign. then again, the Fourier Transforms the same sign into a ‘jw’ plane. also, the Fourier transform is the subset of Laplace rework that has a real element as ‘0’.
Yes, it is able to be. if you have a feature f(t)=zero, then its Laplace rework F(s)=0 may be determined by multiplying the Laplace pair with a scalar 0, which defines the linearity assets.
