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Limit Calculator

Laplace Transform Calculator

Enter the differential equation to find its Laplace transformation.

Equation: Hint: Please write e^(3t) as e^{3t}

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Laplace Transform Calculator:

Use this 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.

What Is Laplace Transform?

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;

  • f(t)= Time function that is defined for the interval t≥0
  • s=Complex Variable (s=a+b𝜄, where ‘a’ is the real and ‘b’ is the imaginary part)
  • \(\int_{0}^{∞}\) = Improper integral of the function
  • F(s) = Function in frequency domain

How To Find Laplace Transform of a Function?

There are two possible methods to find Laplace transform:

1: Using Laplace Formula:

\(F(s) = \int_{0}^{∞} f(t)e^{-st} dt\)

Example:

Suppose we have the function \(f(t)=6e^{-5t}+e^{3t}+5t^{3}-9\)

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

  1. \(6e^{-5t}\):

\(\int_{0}^{∞} 6e^{-5t} e^{-st} dt\)

\(=6\int_{0}^{∞} e^{-(5+s)t} dt\)

\(=6[\dfrac{1}{-(5+s)}e^{-(5+s)t}]_{0}^{∞}\)

\(=6[\dfrac{1}{s+5}]\)

  1. \(e^{3t}\);

\(\int_{0}^{∞} e^{3t} e^{-st} dt\)

\(=\int_{0}^{∞} e^{(3-s)t}dt\)

\(=[\dfrac{1}{3-s}e^{(3-s)t}]_{0}^{∞}\)

\(=\dfrac{1}{s-3}\)

  1. \(5t^{3}\):

\(\int_{0}^{∞} 5t^{3} e^{-st} dt\)

\(=5\int_{0}^{∞}t^{3}e^{-st}dt\)

\(=[\dfrac{1}{3-s}e^{(3-s)t}]_{0}^{∞}\)

\(=5*\dfrac{3!}{s^{4}}\)

\(=\dfrac{30}{s^{4}}\)

  1. -9:

\(\int_{0}^{∞} -9 e^{-st} dt\)

\(=-9\int_{0}^{∞} e^{-st} dt\)

\(=-9[\dfrac{-1}{s}e^{-st}]_{0}^{∞}\)

\(=\dfrac{9}{s}\)

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.

2: Laplace Transform Calculator:

  • Just enter the given function f(t)
  • Click ‘Calculate’
  • Get the frequency domain function F(s)

Laplace Transform Table:

The following Laplace transform table helps you to find the Laplace Transformation of simple and most common functions.

Function name Time-domain function Laplace transform 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

Properties of Laplace Transform:

Our Laplace transform calculator automates the transformations of functions based on the following properties:

Properties of Laplace Transform
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(∞)

Applications:

Laplace calculator has vast applications in various fields including:

  1. To convert time-domain signals into frequency-domain signals that let engineers design responsive control systems.
  2. Students can use the calculator to verify their calculations for complicated Laplace transform problems
  3. It helps to simplify partial differential equations in discrete calculus
  4. Statisticians can benefit from it by deriving the moment-generating function of any probability distribution scenario.
  5. To analyze the behavior of various electrical circuits and check their stability response.

Faqs:

What is the difference between the Fourier and the Laplace Transform?

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’.

Can the Laplace transform equal 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.

Reference: 

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.