The Laplace Transform - Differential Equations

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Question

Find the Laplace Transform for the following function.

$\mathcal{L}\begin{Bmatrix} $e^{-5t}$ \end{Bmatrix}$

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Answer

To find the Laplace Transform for the following function

$\mathcal{L}\begin{Bmatrix} $e^{-5t}$\end{Bmatrix}$

use the transforms for basic functions which state,

$\mathcal{L}\begin{Bmatrix} 1 \end{Bmatrix}=\frac{1}{s}$$

$\mathcal{L}\begin{Bmatrix} $e^{at}$ \end{Bmatrix}=\frac{1}{s-a}$$

$\mathcal{L}\begin{Bmatrix} $e^{-5t}$ \end{Bmatrix}=\int_0^\infty $e^{-st}$$e^{-5t}$dt=\int_0^\infty $e^{-(s+5)t}$ dt$

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Question

Find the Laplace Transform for the following function.

$\mathcal{L}\begin{Bmatrix} $e^{-5t}$ \end{Bmatrix}$

Tap to reveal answer

Answer

To find the Laplace Transform for the following function

$\mathcal{L}\begin{Bmatrix} $e^{-5t}$\end{Bmatrix}$

use the transforms for basic functions which state,

$\mathcal{L}\begin{Bmatrix} 1 \end{Bmatrix}=\frac{1}{s}$$

$\mathcal{L}\begin{Bmatrix} $e^{at}$ \end{Bmatrix}=\frac{1}{s-a}$$

$\mathcal{L}\begin{Bmatrix} $e^{-5t}$ \end{Bmatrix}=\int_0^\infty $e^{-st}$$e^{-5t}$dt=\int_0^\infty $e^{-(s+5)t}$ dt$

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Question

Find the Laplace Transform for the following function.

$\mathcal{L}\begin{Bmatrix} $e^{-5t}$ \end{Bmatrix}$

Tap to reveal answer

Answer

To find the Laplace Transform for the following function

$\mathcal{L}\begin{Bmatrix} $e^{-5t}$\end{Bmatrix}$

use the transforms for basic functions which state,

$\mathcal{L}\begin{Bmatrix} 1 \end{Bmatrix}=\frac{1}{s}$$

$\mathcal{L}\begin{Bmatrix} $e^{at}$ \end{Bmatrix}=\frac{1}{s-a}$$

$\mathcal{L}\begin{Bmatrix} $e^{-5t}$ \end{Bmatrix}=\int_0^\infty $e^{-st}$$e^{-5t}$dt=\int_0^\infty $e^{-(s+5)t}$ dt$

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Question

Find the Laplace Transform for the following function.

$\mathcal{L}\begin{Bmatrix} $e^{-5t}$ \end{Bmatrix}$

Tap to reveal answer

Answer

To find the Laplace Transform for the following function

$\mathcal{L}\begin{Bmatrix} $e^{-5t}$\end{Bmatrix}$

use the transforms for basic functions which state,

$\mathcal{L}\begin{Bmatrix} 1 \end{Bmatrix}=\frac{1}{s}$$

$\mathcal{L}\begin{Bmatrix} $e^{at}$ \end{Bmatrix}=\frac{1}{s-a}$$

$\mathcal{L}\begin{Bmatrix} $e^{-5t}$ \end{Bmatrix}=\int_0^\infty $e^{-st}$$e^{-5t}$dt=\int_0^\infty $e^{-(s+5)t}$ dt$

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Question

Find the Laplace transform of the periodic function.

$S(t)=\left\{\begin{matrix} 1,& 0\leq2\ 0, & 2\leq4 \end{matrix}\right.$

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Answer

This particular piecewise function is called a square wave. The period of this function is the length at which it takes the function to return to its starting point.

For this particular function

$S(t)=\left\{\begin{matrix} 1,& 0\leq2\ 0, & 2\leq4 \end{matrix}\right.$

it has a period of

.

and furthermore,

Using the Transform of a Periodic Function Theorem which states,

$\mathcal{L}\begin{Bmatrix} f(t) $\end{Bmatrix}=\frac{1}{1-e^{-sT}$$}\int_$0^T$ $e^{-sT}$f(t)dt$

the problem can be solved as follows.

$\mathcal{L}\begin{Bmatrix} S(t) $\end{Bmatrix}=\frac{1}{1-e^{-4s}$$}\int_$0^4$ $e^{-st}$S(t)dt$

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Question

Find the Laplace transform of the periodic function.

$S(t)=\left\{\begin{matrix} 1,& 0\leq2\ 0, & 2\leq4 \end{matrix}\right.$

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Answer

This particular piecewise function is called a square wave. The period of this function is the length at which it takes the function to return to its starting point.

For this particular function

$S(t)=\left\{\begin{matrix} 1,& 0\leq2\ 0, & 2\leq4 \end{matrix}\right.$

it has a period of

.

and furthermore,

Using the Transform of a Periodic Function Theorem which states,

$\mathcal{L}\begin{Bmatrix} f(t) $\end{Bmatrix}=\frac{1}{1-e^{-sT}$$}\int_$0^T$ $e^{-sT}$f(t)dt$

the problem can be solved as follows.

$\mathcal{L}\begin{Bmatrix} S(t) $\end{Bmatrix}=\frac{1}{1-e^{-4s}$$}\int_$0^4$ $e^{-st}$S(t)dt$

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Question

Find the Laplace transform of the periodic function.

$S(t)=\left\{\begin{matrix} 1,& 0\leq2\ 0, & 2\leq4 \end{matrix}\right.$

Tap to reveal answer

Answer

This particular piecewise function is called a square wave. The period of this function is the length at which it takes the function to return to its starting point.

For this particular function

$S(t)=\left\{\begin{matrix} 1,& 0\leq2\ 0, & 2\leq4 \end{matrix}\right.$

it has a period of

.

and furthermore,

Using the Transform of a Periodic Function Theorem which states,

$\mathcal{L}\begin{Bmatrix} f(t) $\end{Bmatrix}=\frac{1}{1-e^{-sT}$$}\int_$0^T$ $e^{-sT}$f(t)dt$

the problem can be solved as follows.

$\mathcal{L}\begin{Bmatrix} S(t) $\end{Bmatrix}=\frac{1}{1-e^{-4s}$$}\int_$0^4$ $e^{-st}$S(t)dt$

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Question

Find the Laplace transform of the periodic function.

$S(t)=\left\{\begin{matrix} 1,& 0\leq2\ 0, & 2\leq4 \end{matrix}\right.$

Tap to reveal answer

Answer

This particular piecewise function is called a square wave. The period of this function is the length at which it takes the function to return to its starting point.

For this particular function

$S(t)=\left\{\begin{matrix} 1,& 0\leq2\ 0, & 2\leq4 \end{matrix}\right.$

it has a period of

.

and furthermore,

Using the Transform of a Periodic Function Theorem which states,

$\mathcal{L}\begin{Bmatrix} f(t) $\end{Bmatrix}=\frac{1}{1-e^{-sT}$$}\int_$0^T$ $e^{-sT}$f(t)dt$

the problem can be solved as follows.

$\mathcal{L}\begin{Bmatrix} S(t) $\end{Bmatrix}=\frac{1}{1-e^{-4s}$$}\int_$0^4$ $e^{-st}$S(t)dt$

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The Laplace Transform - Differential Equations

Find the Laplace Transform for the following function.

To find the Laplace Transform for the following function use the transforms for basic functions which state,

Find the Laplace Transform for the following function.

To find the Laplace Transform for the following function use the transforms for basic functions which state,

Find the Laplace Transform for the following function.

To find the Laplace Transform for the following function use the transforms for basic functions which state,

Find the Laplace Transform for the following function.

To find the Laplace Transform for the following function use the transforms for basic functions which state,

Find the Laplace transform of the periodic function.

Find the Laplace transform of the periodic function.

Find the Laplace transform of the periodic function.

Find the Laplace transform of the periodic function.

Find the Laplace Transform for the following function.

$\mathcal{L}\begin{Bmatrix} $e^{-5t}$ \end{Bmatrix}$

Find the Laplace Transform for the following function.

$\mathcal{L}\begin{Bmatrix} $e^{-5t}$ \end{Bmatrix}$

Find the Laplace Transform for the following function.

$\mathcal{L}\begin{Bmatrix} $e^{-5t}$ \end{Bmatrix}$

Find the Laplace Transform for the following function.

$\mathcal{L}\begin{Bmatrix} $e^{-5t}$ \end{Bmatrix}$

Find the Laplace transform of the periodic function.

$S(t)=\left\{\begin{matrix} 1,& 0\leq2\ 0, & 2\leq4 \end{matrix}\right.$

Find the Laplace transform of the periodic function.

$S(t)=\left\{\begin{matrix} 1,& 0\leq2\ 0, & 2\leq4 \end{matrix}\right.$

Find the Laplace transform of the periodic function.

$S(t)=\left\{\begin{matrix} 1,& 0\leq2\ 0, & 2\leq4 \end{matrix}\right.$

Find the Laplace transform of the periodic function.

$S(t)=\left\{\begin{matrix} 1,& 0\leq2\ 0, & 2\leq4 \end{matrix}\right.$