Laplace Transform Calculator

This Laplace Transform Calculator converts a function of t into a function of s. Enter f(t) such as t^2, e^(3t), or sin(2t), and it returns the transform F(s).

Step-by-step method

1. Write the function f(t).
2. Apply the Laplace transform definition (or a standard transform pair).
3. Write the transform F(s).

Formula:

Example 1:

Step 1 - Write the function f(t).

In this problem: We transform \(f(t) = t^{2}\).

Step 2 - Apply the Laplace transform definition (or a standard transform pair).

In this problem: Apply the Laplace transform to the function of t.

Step 3 - Write the transform F(s).

In this problem: The Laplace transform is \(F(s) = \frac{2}{s^{3}}\).

Final answer:

Example 2:

Step 1 - Write the function f(t).

In this problem: We transform \(f(t) = e^{3 t}\).

Step 2 - Apply the Laplace transform definition (or a standard transform pair).

In this problem: Apply the Laplace transform to the function of t.

Step 3 - Write the transform F(s).

In this problem: The Laplace transform is \(F(s) = \frac{1}{s - 3}\).

Final answer:

Example 3:

Step 1 - Write the function f(t).

In this problem: We transform \(f(t) = \sin{\left(2 t \right)}\).

Step 2 - Apply the Laplace transform definition (or a standard transform pair).

In this problem: Apply the Laplace transform to the function of t.

Step 3 - Write the transform F(s).

In this problem: The Laplace transform is \(F(s) = \frac{2}{s^{2} + 4}\).

Final answer:

See Example 1 Hide Example 1

Problem

\(\mathcal{L}\{t^{2}\}\)

Approach

Step-by-step method

1. Write the function f(t).
2. Apply the Laplace transform definition (or a standard transform pair).
3. Write the transform F(s).

Step 1

Step 1 - Write the function f(t).

In this problem: We transform \(f(t) = t^{2}\).

\(f(t) = t^{2}\)

Step 2

Step 2 - Apply the Laplace transform definition (or a standard transform pair).

In this problem: Apply the Laplace transform to the function of t.

\(\mathcal{L}\{f(t)\} = F(s) = \int_{0}^{\infty} e^{-st} f(t) \, dt\)

Step 3

Step 3 - Write the transform F(s).

In this problem: The Laplace transform is \(F(s) = \frac{2}{s^{3}}\).

\(\mathcal{L}\{t^{2}\} = \frac{2}{s^{3}}\)

Final Answer

\(\)

See Example 2 Hide Example 2

Problem

\(\mathcal{L}\{e^{3 t}\}\)

Approach

Step-by-step method

1. Write the function f(t).
2. Apply the Laplace transform definition (or a standard transform pair).
3. Write the transform F(s).

Step 1

Step 1 - Write the function f(t).

In this problem: We transform \(f(t) = e^{3 t}\).

\(f(t) = e^{3 t}\)

Step 2

Step 2 - Apply the Laplace transform definition (or a standard transform pair).

In this problem: Apply the Laplace transform to the function of t.

\(\mathcal{L}\{f(t)\} = F(s) = \int_{0}^{\infty} e^{-st} f(t) \, dt\)

Step 3

Step 3 - Write the transform F(s).

In this problem: The Laplace transform is \(F(s) = \frac{1}{s - 3}\).

\(\mathcal{L}\{e^{3 t}\} = \frac{1}{s - 3}\)

Final Answer

\(\)

See Example 3 Hide Example 3

Problem

\(\mathcal{L}\{\sin{\left(2 t \right)}\}\)

Approach

Step-by-step method

1. Write the function f(t).
2. Apply the Laplace transform definition (or a standard transform pair).
3. Write the transform F(s).

Step 1

Step 1 - Write the function f(t).

In this problem: We transform \(f(t) = \sin{\left(2 t \right)}\).

\(f(t) = \sin{\left(2 t \right)}\)

Step 2

Step 2 - Apply the Laplace transform definition (or a standard transform pair).

In this problem: Apply the Laplace transform to the function of t.

\(\mathcal{L}\{f(t)\} = F(s) = \int_{0}^{\infty} e^{-st} f(t) \, dt\)

Step 3

Step 3 - Write the transform F(s).

In this problem: The Laplace transform is \(F(s) = \frac{2}{s^{2} + 4}\).

\(\mathcal{L}\{\sin{\left(2 t \right)}\} = \frac{2}{s^{2} + 4}\)

Final Answer

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