Enter the differential equation to find its Laplace transformation.
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Use this online Laplace transform calculator to find the Laplace transformation of a function f(t). The calculator applies relevant Laplace transform formula and integral operations for the representation.
Laplace transform is an absolute integral transform that helps you to convert a function in real variable (t) to a function in complex variable (s).
Mathematically,
\(F(s) = \int_{0}^{∞} f(t)e^{-st} dt\)
Where;
There are two possible methods to find Laplace transforms:
\(F(s) = \int_{0}^{∞} f(t)e^{-st} dt\)
Example:
Suppose we have the function given below:
\(f(t)=6e^{-5t}+e^{3t}+5t^{3}-9\)
Step 01: Write down the Laplace transform formula
\(F(s) = \int_{0}^{∞} f(t)e^{-st} dt\)
Step 02: Input the given function f(t)
\(F(s) = \int_{0}^{∞} f(t)e^{-st} dt\) \(F(s) = \int_{0}^{∞} (6e^{-5t}+e^{3t}+5t^{3}-9) e^{-st} dt\)
Step 03: Apply the formula to individual terms
\(\int_{0}^{∞} 6e^{-5t} e^{-st} dt\) \(=6\int_{0}^{∞} e^{-(5+s)t} dt\) \(=6[\dfrac{1}{-(5+s)}e^{-(5+s)t}]_{0}^{∞}\) \(=6[\dfrac{1}{s+5}]\)
\(\int_{0}^{∞} e^{3t} e^{-st} dt\) \(=\int_{0}^{∞} e^{(3-s)t}dt\) \(=[\dfrac{1}{3-s}e^{(3-s)t}]_{0}^{∞}\) \(=\dfrac{1}{s-3}\)
\(\int_{0}^{∞} 5t^{3} e^{-st} dt\) \(=5\int_{0}^{∞}t^{3}e^{-st}dt\) \(=[\dfrac{1}{3-s}e^{(3-s)t}]_{0}^{∞}\) \(=5*\dfrac{3!}{s^{4}}\) \(=\dfrac{30}{s^{4}}\)
\(\int_{0}^{∞} -9 e^{-st} dt\) \(=-9\int_{0}^{∞} e^{-st} dt\) \(=-9[\dfrac{-1}{s}e^{-st}]_{0}^{∞}\) \(=\dfrac{9}{s}\)
Step 04: Sum up all transforms together
\(F(s)=6(\dfrac{1}{s+5}+\dfrac{1}{s-3}+\dfrac{30}{s^{4}}-\dfrac{9}{s}\)
To convert back this function to the original time domain equation, you can use our inverse Laplace transform calculator.
The following Laplace transform table helps you to find the Laplace Transformation of simple and most common functions, provided with the initial conditions:
Function name | Time-domain function | Laplace transforms online |
f (t) | F(s) = L{f (t)} | |
Constant | 1 | 1/s |
Linear | t | 1/\(s^2\) |
Power | t n | n!/\(s^{n+1}\) |
Power | t a | Γ(a+1) ⋅ s -(a+1) |
Exponent | e at | 1/s-a |
Sine | sin at | a/ \(s^2 + a^2\) |
Cosine | cos at | s/ \(s^2 + a^2\) |
Hyperbolic sine | sinh at | a/ \(s^2 - a^2\) |
Hyperbolic cosine | cosh at | s/ \(s^2 - a^2\) |
Growing sine | t sin at | 2as/ \((s^2 + a^2)^2\) |
Growing cosine | t cos at | \(s^2 - a^2/ (s^2 + a^2)^2\) |
Decaying sine | e -at sin ωt | ω /\((s+a)^2 + ω^2\) |
Decaying cosine | e -at cos ωt | (s+a)/\((s+a)^2 + ω^2\) |
Delta function | δ(t) | 1 |
Delayed delta | δ(t-a) | e-as |
Our online Laplace transform calculator automates the transformations of functions based on the following properties:
Property | Equation |
---|---|
Linearity | F(s) = L{f(t)} + L{g(t)} |
Time Delay | L{f(t-td)} = e^(-tsd) F(s) |
First Derivative | L{f'(t)} = sF(s) - f(0-) |
Second Derivative | L{f''(t)} = s^2 F(s) - sf(0-) - f'(0-) |
Nth Derivative | L{f^(n)(t)} = s^n F(s) - s^(n-1)f(0-) - ... - f^(n-1)(0-) |
Integration | L{∫f(t) dt} = 1/s F(s) |
Convolution | L{f(t) * g(t)} = F(s)G(s) |
Initial Value Theorem | lim(s->∞) sF(s) = f(0-) |
Final Value Theorem | lim(s->0) sf(s) = f(∞) |
Laplace calculator has vast applications in various fields including:
The Laplace Transform helps you to convert a simple signal into a complex frequency signal. On the other hand, the Fourier Transforms the same signal into a ‘jw’ plane. Also, the Fourier Transform is the subset of Laplace transform that has a real part as ‘0’.
Yes, it can be. If you have a function f(t)=0, then its Laplace transform F(s)=0 can be determined by multiplying the Laplace pair with a scalar 0, which defines the linearity property.
From the source of Wikipedia: Formal definition, Bilateral Laplace transform, Inverse Laplace transform.
From the source of Paul's Online Notes: Laplace Transforms, Solving IVPs with Laplace Transforms, Nonconstant Coefficient IVP’s.
From the source of Swarth More: Linearity, Time Delay, First Derivative, Second Derivative, Initial Value Theorem, Final Value Theorem.
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