D2 - Laplace transforms from formula and definition


Example 1 πŸ”—

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 πŸ”—

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 πŸ”—

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 πŸ”—

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 πŸ”—

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 πŸ”—

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 πŸ”—

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 πŸ”—

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 πŸ”—

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 πŸ”—

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 πŸ”—

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 πŸ”—

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 πŸ”—

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 πŸ”—

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 πŸ”—

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 πŸ”—

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 πŸ”—

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 πŸ”—

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 πŸ”—

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 πŸ”—

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 πŸ”—

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 πŸ”—

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 πŸ”—

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 πŸ”—

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 πŸ”—

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 πŸ”—

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 πŸ”—

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 πŸ”—

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 πŸ”—

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 πŸ”—

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 πŸ”—

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 πŸ”—

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 πŸ”—

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 πŸ”—

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 πŸ”—

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 πŸ”—

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 πŸ”—

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 πŸ”—

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 πŸ”—

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 πŸ”—

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 πŸ”—

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 πŸ”—

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 πŸ”—

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 πŸ”—

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 πŸ”—

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 πŸ”—

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 πŸ”—

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 πŸ”—

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 πŸ”—

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 πŸ”—

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:

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