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Laplace Transform Calculator

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

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An online Laplace transform calculator will help you to provide the transformation of the real variable function to the complex variable. The Laplace transformation has many applications in engineering and science such as the analysis of control systems and electronic circuit’s etc. Also, the Laplace solver is used for solving differential equations with the help of the Laplace transform equation. Read on to understand how to find Laplace transformations and many more!

What is a Laplace Transform?

In mathematics, Laplace transformations are integral transformations, which change a real variable function f (t) to a complex variable function. The reason behind this transformation is to change ordinary differential equations into the algebraic equation which helps to determine ordinary differential equations.

So, a linear differential equation is extremely prevalent in real-world applications and commonly arises from problems in physics, electrical engineering, and control systems. Apart from this, the Laplace transform calculator can only calculate normal Laplace transform which is a process known as unilateral Laplace transform. This is due to the use of one side of the Laplace transform (normal side) and neglect to use the inverse Laplace transform side.

However, an online Integral Calculator helps you to evaluate the integrals of the functions with respect to the variable involved.

Laplace Transform Formula:

The standard form of unilateral laplace transform equation L is:

$$F(s)=L(f(t))=∫^∞_0e^{−st}f(t)dt$$

Where f(t) is defined as all real numbers \(t ≥ 0\) and (s) is a complex number frequency parameter.

Find Laplace Transform of a Function:

Have a look at the detailed step-wise process that is helpful in computing the Laplace Transform of any equation, if you’re not using the Laplace transform calculator you can do all these calculations manually by following these steps:

  • First, take an equation or a function to which you need to perform the operation.
  • Then, perform the integral operation on the given equation and do all mathematical calculations to the solution.
  • Now, to get the result, substitute all values in the obtained equation.

Example:

Find Laplace Transform of \( 5sinh(4t)+5sin(4t)?\)

Solution:

By taking the functions \(f(t), g(t)\)

$$f(t)=sinh(4t)$$

$$g(t)=sin(4t)$$

Now, Using the linearity property of Laplace

$$L[a.f(t)+b.g(t)]=a.L[f(t)]+b.L[g(t)]$$

$$L[sinh(4t)]=4/s^2-4$$

$$Lsin(2t)]=4/s^2+4$$

$$L[5sinh(4t)+5sin(4t)]=5×4/s^2-4+5×4/s^2+4$$

$$4=20/s^2-4+20/s2+4=20x{1/s^2-4+1/s^2+4}$$

$$=20x{s^2+4+s^2-4/s^4-16}$$

$$=20x{2s^2/s^4-16}$$

$$=40s^2/s^4-16$$

Hence, the Laplace Transform of \(5sinh(4t)+5sin(4t) \text { is} 40s^2/s^4-16\).

Table for Laplace:

In order to perform the Laplace transformations of linear equations the Laplace solver follows the table:

Function name Time-domain function Laplace transform
(t) F(s) = L{(t)}
Constant 1 1/s
Linear t 1/\(s^2\)
Power t n n!/\(s^{n+1}\)
Power t a Γ(a+1) ⋅ -(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

Moreover, an online Derivative Calculator allows you to find the derivative of the function with respect to a given variable.

How Laplace Transform Calculator Works?

An online Laplace transformation calculator helps you to transform real functions into complex function with these steps:

Input:

  • First, enter a simple equation, and you can see the equation preview.
  • Hit the calculate button for further process.

Output:

The Laplace transform calculator displays the following results:

  • First of all, the calculator shows your input in the form of the ordinary differential equation.
  • Then, provide the answer against the equation in algebraic form.

FAQs for Laplace Transform:

What are the Main Applications of Laplace Transform?

The main applications of Laplace transform are:

  • Signal Process
  • Derive the circuit
  • Control system
  • Integrated Circuit

Define the Difference Between Fourier and Laplace Transform?

The Fourier transform decomposes a function that depends on space or time, changing the magnitudes of a signal. On the other hand, the Laplace transform changes the oscillation and magnitude parts. Actually, the Laplace Transform is the main set of the Fourier Transform.

What is the Significance of Laplace Transform?

The Laplace transform has useful techniques for finding certain verities of differential equations when primary conditions are available, especially when the initial values are zero.

Final Words:

Use this handy Laplace transform calculator that displays the transformation of a real ordinary differential equation into a complex algebraic function. Undoubtedly, you can do all these calculations manually but it’s a lengthy and time taking process. So, with the help of this Laplace transformation calculator students and professionals make the calculation instantly.

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.