## D2 - Laplace transforms from formula and definition

#### Example 1 π

Compute the Laplace transform $$\mathcal{L}\{y\}$$ of $$y = 3 \, \delta\left(t - 4\right) - 2 \, e^{\left(2 \, t\right)} - 4 \, \mathrm{u}\left(t - 2\right)$$ by using a transform table.

Then show how the integral definition of the Laplace transform to obtains same result.

Answer:

$\mathcal{L}\{y\} = -\frac{4 \, e^{\left(-2 \, s\right)}}{s} - \frac{2}{s - 2} + 3 \, e^{\left(-4 \, s\right)}$

#### Example 2 π

Compute the Laplace transform $$\mathcal{L}\{y\}$$ of $$y = -4 \, \delta\left(t - 2\right) - 5 \, e^{\left(2 \, t\right)} - 3 \, \mathrm{u}\left(t - 4\right)$$ by using a transform table.

Then show how the integral definition of the Laplace transform to obtains same result.

Answer:

$\mathcal{L}\{y\} = -\frac{3 \, e^{\left(-4 \, s\right)}}{s} - \frac{5}{s - 2} - 4 \, e^{\left(-2 \, s\right)}$

#### Example 3 π

Compute the Laplace transform $$\mathcal{L}\{y\}$$ of $$y = 3 \, \delta\left(t - 1\right) + 4 \, e^{\left(2 \, t\right)} + 5 \, \mathrm{u}\left(t - 4\right)$$ by using a transform table.

Then show how the integral definition of the Laplace transform to obtains same result.

Answer:

$\mathcal{L}\{y\} = \frac{5 \, e^{\left(-4 \, s\right)}}{s} + \frac{4}{s - 2} + 3 \, e^{\left(-s\right)}$

#### Example 4 π

Compute the Laplace transform $$\mathcal{L}\{y\}$$ of $$y = 5 \, \delta\left(t - 1\right) + 3 \, e^{\left(4 \, t\right)} + 4 \, \mathrm{u}\left(t - 5\right)$$ by using a transform table.

Then show how the integral definition of the Laplace transform to obtains same result.

Answer:

$\mathcal{L}\{y\} = \frac{4 \, e^{\left(-5 \, s\right)}}{s} + \frac{3}{s - 4} + 5 \, e^{\left(-s\right)}$

#### Example 5 π

Compute the Laplace transform $$\mathcal{L}\{y\}$$ of $$y = 3 \, \delta\left(t - 3\right) - 4 \, e^{\left(3 \, t\right)} - 2 \, \mathrm{u}\left(t - 5\right)$$ by using a transform table.

Then show how the integral definition of the Laplace transform to obtains same result.

Answer:

$\mathcal{L}\{y\} = -\frac{2 \, e^{\left(-5 \, s\right)}}{s} - \frac{4}{s - 3} + 3 \, e^{\left(-3 \, s\right)}$

#### Example 6 π

Compute the Laplace transform $$\mathcal{L}\{y\}$$ of $$y = 3 \, \delta\left(t - 4\right) + 4 \, e^{\left(3 \, t\right)} - 4 \, \mathrm{u}\left(t - 1\right)$$ by using a transform table.

Then show how the integral definition of the Laplace transform to obtains same result.

Answer:

$\mathcal{L}\{y\} = -\frac{4 \, e^{\left(-s\right)}}{s} + \frac{4}{s - 3} + 3 \, e^{\left(-4 \, s\right)}$

#### Example 7 π

Compute the Laplace transform $$\mathcal{L}\{y\}$$ of $$y = -4 \, \delta\left(t - 4\right) + 2 \, e^{\left(4 \, t\right)} - 3 \, \mathrm{u}\left(t - 1\right)$$ by using a transform table.

Then show how the integral definition of the Laplace transform to obtains same result.

Answer:

$\mathcal{L}\{y\} = -\frac{3 \, e^{\left(-s\right)}}{s} + \frac{2}{s - 4} - 4 \, e^{\left(-4 \, s\right)}$

#### Example 8 π

Compute the Laplace transform $$\mathcal{L}\{y\}$$ of $$y = 5 \, \delta\left(t - 1\right) - 2 \, e^{\left(2 \, t\right)} + 5 \, \mathrm{u}\left(t - 1\right)$$ by using a transform table.

Then show how the integral definition of the Laplace transform to obtains same result.

Answer:

$\mathcal{L}\{y\} = \frac{5 \, e^{\left(-s\right)}}{s} - \frac{2}{s - 2} + 5 \, e^{\left(-s\right)}$

#### Example 9 π

Compute the Laplace transform $$\mathcal{L}\{y\}$$ of $$y = -3 \, \delta\left(t - 1\right) - 4 \, e^{\left(3 \, t\right)} + 4 \, \mathrm{u}\left(t - 3\right)$$ by using a transform table.

Then show how the integral definition of the Laplace transform to obtains same result.

Answer:

$\mathcal{L}\{y\} = \frac{4 \, e^{\left(-3 \, s\right)}}{s} - \frac{4}{s - 3} - 3 \, e^{\left(-s\right)}$

#### Example 10 π

Compute the Laplace transform $$\mathcal{L}\{y\}$$ of $$y = -5 \, \delta\left(t - 5\right) + 2 \, e^{\left(4 \, t\right)} - 4 \, \mathrm{u}\left(t - 1\right)$$ by using a transform table.

Then show how the integral definition of the Laplace transform to obtains same result.

Answer:

$\mathcal{L}\{y\} = -\frac{4 \, e^{\left(-s\right)}}{s} + \frac{2}{s - 4} - 5 \, e^{\left(-5 \, s\right)}$

#### Example 11 π

Compute the Laplace transform $$\mathcal{L}\{y\}$$ of $$y = -2 \, \delta\left(t - 1\right) - 5 \, e^{\left(5 \, t\right)} - 5 \, \mathrm{u}\left(t - 3\right)$$ by using a transform table.

Then show how the integral definition of the Laplace transform to obtains same result.

Answer:

$\mathcal{L}\{y\} = -\frac{5 \, e^{\left(-3 \, s\right)}}{s} - \frac{5}{s - 5} - 2 \, e^{\left(-s\right)}$

#### Example 12 π

Compute the Laplace transform $$\mathcal{L}\{y\}$$ of $$y = -5 \, \delta\left(t - 1\right) + 2 \, e^{\left(4 \, t\right)} + 2 \, \mathrm{u}\left(t - 2\right)$$ by using a transform table.

Then show how the integral definition of the Laplace transform to obtains same result.

Answer:

$\mathcal{L}\{y\} = \frac{2 \, e^{\left(-2 \, s\right)}}{s} + \frac{2}{s - 4} - 5 \, e^{\left(-s\right)}$

#### Example 13 π

Compute the Laplace transform $$\mathcal{L}\{y\}$$ of $$y = 3 \, \delta\left(t - 4\right) + 4 \, e^{\left(2 \, t\right)} + 5 \, \mathrm{u}\left(t - 4\right)$$ by using a transform table.

Then show how the integral definition of the Laplace transform to obtains same result.

Answer:

$\mathcal{L}\{y\} = \frac{5 \, e^{\left(-4 \, s\right)}}{s} + \frac{4}{s - 2} + 3 \, e^{\left(-4 \, s\right)}$

#### Example 14 π

Compute the Laplace transform $$\mathcal{L}\{y\}$$ of $$y = -5 \, \delta\left(t - 1\right) - 5 \, e^{\left(2 \, t\right)} + 5 \, \mathrm{u}\left(t - 5\right)$$ by using a transform table.

Then show how the integral definition of the Laplace transform to obtains same result.

Answer:

$\mathcal{L}\{y\} = \frac{5 \, e^{\left(-5 \, s\right)}}{s} - \frac{5}{s - 2} - 5 \, e^{\left(-s\right)}$

#### Example 15 π

Compute the Laplace transform $$\mathcal{L}\{y\}$$ of $$y = -4 \, \delta\left(t - 1\right) + 3 \, e^{\left(4 \, t\right)} + 3 \, \mathrm{u}\left(t - 4\right)$$ by using a transform table.

Then show how the integral definition of the Laplace transform to obtains same result.

Answer:

$\mathcal{L}\{y\} = \frac{3 \, e^{\left(-4 \, s\right)}}{s} + \frac{3}{s - 4} - 4 \, e^{\left(-s\right)}$

#### Example 16 π

Compute the Laplace transform $$\mathcal{L}\{y\}$$ of $$y = 3 \, \delta\left(t - 1\right) + 3 \, e^{\left(2 \, t\right)} - 5 \, \mathrm{u}\left(t - 2\right)$$ by using a transform table.

Then show how the integral definition of the Laplace transform to obtains same result.

Answer:

$\mathcal{L}\{y\} = -\frac{5 \, e^{\left(-2 \, s\right)}}{s} + \frac{3}{s - 2} + 3 \, e^{\left(-s\right)}$

#### Example 17 π

Compute the Laplace transform $$\mathcal{L}\{y\}$$ of $$y = 2 \, \delta\left(t - 5\right) - 5 \, e^{\left(2 \, t\right)} - 4 \, \mathrm{u}\left(t - 3\right)$$ by using a transform table.

Then show how the integral definition of the Laplace transform to obtains same result.

Answer:

$\mathcal{L}\{y\} = -\frac{4 \, e^{\left(-3 \, s\right)}}{s} - \frac{5}{s - 2} + 2 \, e^{\left(-5 \, s\right)}$

#### Example 18 π

Compute the Laplace transform $$\mathcal{L}\{y\}$$ of $$y = -5 \, \delta\left(t - 3\right) - 4 \, e^{\left(5 \, t\right)} - 3 \, \mathrm{u}\left(t - 3\right)$$ by using a transform table.

Then show how the integral definition of the Laplace transform to obtains same result.

Answer:

$\mathcal{L}\{y\} = -\frac{3 \, e^{\left(-3 \, s\right)}}{s} - \frac{4}{s - 5} - 5 \, e^{\left(-3 \, s\right)}$

#### Example 19 π

Compute the Laplace transform $$\mathcal{L}\{y\}$$ of $$y = 4 \, \delta\left(t - 1\right) - 5 \, e^{\left(4 \, t\right)} + 2 \, \mathrm{u}\left(t - 5\right)$$ by using a transform table.

Then show how the integral definition of the Laplace transform to obtains same result.

Answer:

$\mathcal{L}\{y\} = \frac{2 \, e^{\left(-5 \, s\right)}}{s} - \frac{5}{s - 4} + 4 \, e^{\left(-s\right)}$

#### Example 20 π

Compute the Laplace transform $$\mathcal{L}\{y\}$$ of $$y = -2 \, \delta\left(t - 2\right) - 5 \, e^{\left(3 \, t\right)} - 2 \, \mathrm{u}\left(t - 4\right)$$ by using a transform table.

Then show how the integral definition of the Laplace transform to obtains same result.

Answer:

$\mathcal{L}\{y\} = -\frac{2 \, e^{\left(-4 \, s\right)}}{s} - \frac{5}{s - 3} - 2 \, e^{\left(-2 \, s\right)}$

#### Example 21 π

Compute the Laplace transform $$\mathcal{L}\{y\}$$ of $$y = -3 \, \delta\left(t - 4\right) + 5 \, e^{\left(2 \, t\right)} - 3 \, \mathrm{u}\left(t - 5\right)$$ by using a transform table.

Then show how the integral definition of the Laplace transform to obtains same result.

Answer:

$\mathcal{L}\{y\} = -\frac{3 \, e^{\left(-5 \, s\right)}}{s} + \frac{5}{s - 2} - 3 \, e^{\left(-4 \, s\right)}$

#### Example 22 π

Compute the Laplace transform $$\mathcal{L}\{y\}$$ of $$y = 3 \, \delta\left(t - 1\right) - 3 \, e^{\left(2 \, t\right)} + 5 \, \mathrm{u}\left(t - 5\right)$$ by using a transform table.

Then show how the integral definition of the Laplace transform to obtains same result.

Answer:

$\mathcal{L}\{y\} = \frac{5 \, e^{\left(-5 \, s\right)}}{s} - \frac{3}{s - 2} + 3 \, e^{\left(-s\right)}$

#### Example 23 π

Compute the Laplace transform $$\mathcal{L}\{y\}$$ of $$y = -5 \, \delta\left(t - 2\right) + 3 \, e^{\left(5 \, t\right)} + 2 \, \mathrm{u}\left(t - 5\right)$$ by using a transform table.

Then show how the integral definition of the Laplace transform to obtains same result.

Answer:

$\mathcal{L}\{y\} = \frac{2 \, e^{\left(-5 \, s\right)}}{s} + \frac{3}{s - 5} - 5 \, e^{\left(-2 \, s\right)}$

#### Example 24 π

Compute the Laplace transform $$\mathcal{L}\{y\}$$ of $$y = -5 \, \delta\left(t - 1\right) - 4 \, e^{\left(4 \, t\right)} - 3 \, \mathrm{u}\left(t - 4\right)$$ by using a transform table.

Then show how the integral definition of the Laplace transform to obtains same result.

Answer:

$\mathcal{L}\{y\} = -\frac{3 \, e^{\left(-4 \, s\right)}}{s} - \frac{4}{s - 4} - 5 \, e^{\left(-s\right)}$

#### Example 25 π

Compute the Laplace transform $$\mathcal{L}\{y\}$$ of $$y = -3 \, \delta\left(t - 3\right) + 4 \, e^{\left(5 \, t\right)} - 2 \, \mathrm{u}\left(t - 2\right)$$ by using a transform table.

Then show how the integral definition of the Laplace transform to obtains same result.

Answer:

$\mathcal{L}\{y\} = -\frac{2 \, e^{\left(-2 \, s\right)}}{s} + \frac{4}{s - 5} - 3 \, e^{\left(-3 \, s\right)}$

#### Example 26 π

Compute the Laplace transform $$\mathcal{L}\{y\}$$ of $$y = -2 \, \delta\left(t - 3\right) + 5 \, e^{\left(3 \, t\right)} + 4 \, \mathrm{u}\left(t - 5\right)$$ by using a transform table.

Then show how the integral definition of the Laplace transform to obtains same result.

Answer:

$\mathcal{L}\{y\} = \frac{4 \, e^{\left(-5 \, s\right)}}{s} + \frac{5}{s - 3} - 2 \, e^{\left(-3 \, s\right)}$

#### Example 27 π

Compute the Laplace transform $$\mathcal{L}\{y\}$$ of $$y = 3 \, \delta\left(t - 4\right) - 3 \, e^{\left(4 \, t\right)} - 4 \, \mathrm{u}\left(t - 4\right)$$ by using a transform table.

Then show how the integral definition of the Laplace transform to obtains same result.

Answer:

$\mathcal{L}\{y\} = -\frac{4 \, e^{\left(-4 \, s\right)}}{s} - \frac{3}{s - 4} + 3 \, e^{\left(-4 \, s\right)}$

#### Example 28 π

Compute the Laplace transform $$\mathcal{L}\{y\}$$ of $$y = -5 \, \delta\left(t - 4\right) - 4 \, e^{\left(4 \, t\right)} + 2 \, \mathrm{u}\left(t - 1\right)$$ by using a transform table.

Then show how the integral definition of the Laplace transform to obtains same result.

Answer:

$\mathcal{L}\{y\} = \frac{2 \, e^{\left(-s\right)}}{s} - \frac{4}{s - 4} - 5 \, e^{\left(-4 \, s\right)}$

#### Example 29 π

Compute the Laplace transform $$\mathcal{L}\{y\}$$ of $$y = 5 \, \delta\left(t - 1\right) + 4 \, e^{\left(2 \, t\right)} - 3 \, \mathrm{u}\left(t - 1\right)$$ by using a transform table.

Then show how the integral definition of the Laplace transform to obtains same result.

Answer:

$\mathcal{L}\{y\} = -\frac{3 \, e^{\left(-s\right)}}{s} + \frac{4}{s - 2} + 5 \, e^{\left(-s\right)}$

#### Example 30 π

Compute the Laplace transform $$\mathcal{L}\{y\}$$ of $$y = 5 \, \delta\left(t - 4\right) + 2 \, e^{\left(3 \, t\right)} - 3 \, \mathrm{u}\left(t - 2\right)$$ by using a transform table.

Then show how the integral definition of the Laplace transform to obtains same result.

Answer:

$\mathcal{L}\{y\} = -\frac{3 \, e^{\left(-2 \, s\right)}}{s} + \frac{2}{s - 3} + 5 \, e^{\left(-4 \, s\right)}$

#### Example 31 π

Compute the Laplace transform $$\mathcal{L}\{y\}$$ of $$y = 3 \, \delta\left(t - 5\right) - 3 \, e^{\left(3 \, t\right)} + 2 \, \mathrm{u}\left(t - 3\right)$$ by using a transform table.

Then show how the integral definition of the Laplace transform to obtains same result.

Answer:

$\mathcal{L}\{y\} = \frac{2 \, e^{\left(-3 \, s\right)}}{s} - \frac{3}{s - 3} + 3 \, e^{\left(-5 \, s\right)}$

#### Example 32 π

Compute the Laplace transform $$\mathcal{L}\{y\}$$ of $$y = 5 \, \delta\left(t - 4\right) - 4 \, e^{\left(5 \, t\right)} - 4 \, \mathrm{u}\left(t - 1\right)$$ by using a transform table.

Then show how the integral definition of the Laplace transform to obtains same result.

Answer:

$\mathcal{L}\{y\} = -\frac{4 \, e^{\left(-s\right)}}{s} - \frac{4}{s - 5} + 5 \, e^{\left(-4 \, s\right)}$

#### Example 33 π

Compute the Laplace transform $$\mathcal{L}\{y\}$$ of $$y = -4 \, \delta\left(t - 5\right) + 2 \, e^{\left(2 \, t\right)} - 2 \, \mathrm{u}\left(t - 4\right)$$ by using a transform table.

Then show how the integral definition of the Laplace transform to obtains same result.

Answer:

$\mathcal{L}\{y\} = -\frac{2 \, e^{\left(-4 \, s\right)}}{s} + \frac{2}{s - 2} - 4 \, e^{\left(-5 \, s\right)}$

#### Example 34 π

Compute the Laplace transform $$\mathcal{L}\{y\}$$ of $$y = -3 \, \delta\left(t - 1\right) - 5 \, e^{\left(4 \, t\right)} - 4 \, \mathrm{u}\left(t - 5\right)$$ by using a transform table.

Then show how the integral definition of the Laplace transform to obtains same result.

Answer:

$\mathcal{L}\{y\} = -\frac{4 \, e^{\left(-5 \, s\right)}}{s} - \frac{5}{s - 4} - 3 \, e^{\left(-s\right)}$

#### Example 35 π

Compute the Laplace transform $$\mathcal{L}\{y\}$$ of $$y = 4 \, \delta\left(t - 5\right) - 4 \, e^{\left(5 \, t\right)} + 5 \, \mathrm{u}\left(t - 3\right)$$ by using a transform table.

Then show how the integral definition of the Laplace transform to obtains same result.

Answer:

$\mathcal{L}\{y\} = \frac{5 \, e^{\left(-3 \, s\right)}}{s} - \frac{4}{s - 5} + 4 \, e^{\left(-5 \, s\right)}$

#### Example 36 π

Compute the Laplace transform $$\mathcal{L}\{y\}$$ of $$y = 2 \, \delta\left(t - 2\right) - 5 \, e^{\left(5 \, t\right)} + 4 \, \mathrm{u}\left(t - 1\right)$$ by using a transform table.

Then show how the integral definition of the Laplace transform to obtains same result.

Answer:

$\mathcal{L}\{y\} = \frac{4 \, e^{\left(-s\right)}}{s} - \frac{5}{s - 5} + 2 \, e^{\left(-2 \, s\right)}$

#### Example 37 π

Compute the Laplace transform $$\mathcal{L}\{y\}$$ of $$y = 3 \, \delta\left(t - 3\right) - 5 \, e^{\left(5 \, t\right)} - 2 \, \mathrm{u}\left(t - 3\right)$$ by using a transform table.

Then show how the integral definition of the Laplace transform to obtains same result.

Answer:

$\mathcal{L}\{y\} = -\frac{2 \, e^{\left(-3 \, s\right)}}{s} - \frac{5}{s - 5} + 3 \, e^{\left(-3 \, s\right)}$

#### Example 38 π

Compute the Laplace transform $$\mathcal{L}\{y\}$$ of $$y = 4 \, \delta\left(t - 1\right) - 5 \, e^{\left(3 \, t\right)} + 2 \, \mathrm{u}\left(t - 3\right)$$ by using a transform table.

Then show how the integral definition of the Laplace transform to obtains same result.

Answer:

$\mathcal{L}\{y\} = \frac{2 \, e^{\left(-3 \, s\right)}}{s} - \frac{5}{s - 3} + 4 \, e^{\left(-s\right)}$

#### Example 39 π

Compute the Laplace transform $$\mathcal{L}\{y\}$$ of $$y = 2 \, \delta\left(t - 3\right) - 5 \, e^{\left(5 \, t\right)} - 5 \, \mathrm{u}\left(t - 3\right)$$ by using a transform table.

Then show how the integral definition of the Laplace transform to obtains same result.

Answer:

$\mathcal{L}\{y\} = -\frac{5 \, e^{\left(-3 \, s\right)}}{s} - \frac{5}{s - 5} + 2 \, e^{\left(-3 \, s\right)}$

#### Example 40 π

Compute the Laplace transform $$\mathcal{L}\{y\}$$ of $$y = -4 \, \delta\left(t - 1\right) - 4 \, e^{\left(2 \, t\right)} - 3 \, \mathrm{u}\left(t - 5\right)$$ by using a transform table.

Then show how the integral definition of the Laplace transform to obtains same result.

Answer:

$\mathcal{L}\{y\} = -\frac{3 \, e^{\left(-5 \, s\right)}}{s} - \frac{4}{s - 2} - 4 \, e^{\left(-s\right)}$

#### Example 41 π

Compute the Laplace transform $$\mathcal{L}\{y\}$$ of $$y = -4 \, \delta\left(t - 2\right) - 4 \, e^{\left(5 \, t\right)} - 4 \, \mathrm{u}\left(t - 2\right)$$ by using a transform table.

Then show how the integral definition of the Laplace transform to obtains same result.

Answer:

$\mathcal{L}\{y\} = -\frac{4 \, e^{\left(-2 \, s\right)}}{s} - \frac{4}{s - 5} - 4 \, e^{\left(-2 \, s\right)}$

#### Example 42 π

Compute the Laplace transform $$\mathcal{L}\{y\}$$ of $$y = -3 \, \delta\left(t - 1\right) + 3 \, e^{\left(2 \, t\right)} + 3 \, \mathrm{u}\left(t - 3\right)$$ by using a transform table.

Then show how the integral definition of the Laplace transform to obtains same result.

Answer:

$\mathcal{L}\{y\} = \frac{3 \, e^{\left(-3 \, s\right)}}{s} + \frac{3}{s - 2} - 3 \, e^{\left(-s\right)}$

#### Example 43 π

Compute the Laplace transform $$\mathcal{L}\{y\}$$ of $$y = -2 \, \delta\left(t - 3\right) + 3 \, e^{\left(5 \, t\right)} - 4 \, \mathrm{u}\left(t - 1\right)$$ by using a transform table.

Then show how the integral definition of the Laplace transform to obtains same result.

Answer:

$\mathcal{L}\{y\} = -\frac{4 \, e^{\left(-s\right)}}{s} + \frac{3}{s - 5} - 2 \, e^{\left(-3 \, s\right)}$

#### Example 44 π

Compute the Laplace transform $$\mathcal{L}\{y\}$$ of $$y = -2 \, \delta\left(t - 2\right) - 2 \, e^{\left(4 \, t\right)} - 4 \, \mathrm{u}\left(t - 5\right)$$ by using a transform table.

Then show how the integral definition of the Laplace transform to obtains same result.

Answer:

$\mathcal{L}\{y\} = -\frac{4 \, e^{\left(-5 \, s\right)}}{s} - \frac{2}{s - 4} - 2 \, e^{\left(-2 \, s\right)}$

#### Example 45 π

Compute the Laplace transform $$\mathcal{L}\{y\}$$ of $$y = 5 \, \delta\left(t - 5\right) + 3 \, e^{\left(3 \, t\right)} - 5 \, \mathrm{u}\left(t - 1\right)$$ by using a transform table.

Then show how the integral definition of the Laplace transform to obtains same result.

Answer:

$\mathcal{L}\{y\} = -\frac{5 \, e^{\left(-s\right)}}{s} + \frac{3}{s - 3} + 5 \, e^{\left(-5 \, s\right)}$

#### Example 46 π

Compute the Laplace transform $$\mathcal{L}\{y\}$$ of $$y = 2 \, \delta\left(t - 2\right) + 3 \, e^{\left(3 \, t\right)} - 3 \, \mathrm{u}\left(t - 1\right)$$ by using a transform table.

Then show how the integral definition of the Laplace transform to obtains same result.

Answer:

$\mathcal{L}\{y\} = -\frac{3 \, e^{\left(-s\right)}}{s} + \frac{3}{s - 3} + 2 \, e^{\left(-2 \, s\right)}$

#### Example 47 π

Compute the Laplace transform $$\mathcal{L}\{y\}$$ of $$y = -3 \, \delta\left(t - 2\right) - 4 \, e^{\left(3 \, t\right)} - 2 \, \mathrm{u}\left(t - 5\right)$$ by using a transform table.

Then show how the integral definition of the Laplace transform to obtains same result.

Answer:

$\mathcal{L}\{y\} = -\frac{2 \, e^{\left(-5 \, s\right)}}{s} - \frac{4}{s - 3} - 3 \, e^{\left(-2 \, s\right)}$

#### Example 48 π

Compute the Laplace transform $$\mathcal{L}\{y\}$$ of $$y = -5 \, \delta\left(t - 3\right) + 4 \, e^{\left(4 \, t\right)} + 4 \, \mathrm{u}\left(t - 5\right)$$ by using a transform table.

Then show how the integral definition of the Laplace transform to obtains same result.

Answer:

$\mathcal{L}\{y\} = \frac{4 \, e^{\left(-5 \, s\right)}}{s} + \frac{4}{s - 4} - 5 \, e^{\left(-3 \, s\right)}$

#### Example 49 π

Compute the Laplace transform $$\mathcal{L}\{y\}$$ of $$y = -5 \, \delta\left(t - 4\right) - 2 \, e^{\left(3 \, t\right)} - 5 \, \mathrm{u}\left(t - 2\right)$$ by using a transform table.

Then show how the integral definition of the Laplace transform to obtains same result.

Answer:

$\mathcal{L}\{y\} = -\frac{5 \, e^{\left(-2 \, s\right)}}{s} - \frac{2}{s - 3} - 5 \, e^{\left(-4 \, s\right)}$

#### Example 50 π

Compute the Laplace transform $$\mathcal{L}\{y\}$$ of $$y = 5 \, \delta\left(t - 4\right) - 2 \, e^{\left(5 \, t\right)} - 2 \, \mathrm{u}\left(t - 1\right)$$ by using a transform table.

Then show how the integral definition of the Laplace transform to obtains same result.

Answer:

$\mathcal{L}\{y\} = -\frac{2 \, e^{\left(-s\right)}}{s} - \frac{2}{s - 5} + 5 \, e^{\left(-4 \, s\right)}$