Free Laplace Transform Calculator With Detailed Steps.

Laplace Transform Calculator:

Use this Laplace transform calculator to find the Laplace transformation of a function f(t) or ordinary differential equation (ODE). Our calculator uses relevant Laplace transform formulas and integral operations to provide precise results with detailed steps.

What Is Laplace Transform?

Laplace transform is a mathematical technique that is used to convert a function f(t) into a function of complex variable(s). This technique is widely used in physics, engineering and control theory to solve ordinary differential equations (ODE'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+b?, 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 transforms:

1 - Using Laplace Formula:

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

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

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

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

\(e^{3t}\);

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

\(5t^{3}\):

\(\int_{0}^{∞} 5t^{3} e^{-st} dt\) \(=5\int_{0}^{∞}t^{3}e^{-st}dt\) \(=[\dfrac{1}{3-s}e^{(3-s)t}]_{0}^{∞}\) \(=5*\dfrac{3!}{s^{4}}\) \(=\dfrac{30}{s^{4}}\)

\(\int_{0}^{∞} -9 e^{-st} dt\) \(=-9\int_{0}^{∞} e^{-st} dt\) \(=-9[\dfrac{-1}{s}e^{-st}]_{0}^{∞}\) \(=\dfrac{9}{s}\)

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 Transformation 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 transforms online
Function name	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 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(∞)

Applications:

Laplace calculator has vast applications in various fields including:

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

FAQ's:

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

Using the Laplace Transform you can convert a time-domain signal into a complex frequency domain signal. On the other hand, the Fourier Transforms convert the same signal into a ‘jw’ (complex frequency) plane. It is also the subset of Laplace transform, especially when the real part of the frequency is ‘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. It means the transformation of a scalar multiple is the scalar multiplication of the transformed function.

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.