## D3 - Inverse Laplace transforms

#### Example 1 π

Explain how to use convolution along with the formula $$\mathcal{L}\{e^{at})=\frac{1}{s-a}$$to find the following inverse Laplace transform:

$\mathcal{L}^{-1}\left\{ \frac{7}{5 \, {\left(s - 5\right)} {\left(s - 7\right)}} \right\}$

$\frac{7}{10} \, e^{\left(7 \, t\right)} - \frac{7}{10} \, e^{\left(5 \, t\right)}$

#### Example 2 π

Explain how to use convolution along with the formula $$\mathcal{L}\{e^{at})=\frac{1}{s-a}$$to find the following inverse Laplace transform:

$\mathcal{L}^{-1}\left\{ \frac{2}{5 \, {\left(s + 5\right)} {\left(s - 6\right)}} \right\}$

$\frac{2}{55} \, e^{\left(6 \, t\right)} - \frac{2}{55} \, e^{\left(-5 \, t\right)}$

#### Example 3 π

Explain how to use convolution along with the formula $$\mathcal{L}\{e^{at})=\frac{1}{s-a}$$to find the following inverse Laplace transform:

$\mathcal{L}^{-1}\left\{ \frac{3}{7 \, {\left(s + 7\right)} {\left(s + 3\right)}} \right\}$

$\frac{3}{28} \, e^{\left(-3 \, t\right)} - \frac{3}{28} \, e^{\left(-7 \, t\right)}$

#### Example 4 π

Explain how to use convolution along with the formula $$\mathcal{L}\{e^{at})=\frac{1}{s-a}$$to find the following inverse Laplace transform:

$\mathcal{L}^{-1}\left\{ \frac{5}{7 \, {\left(s + 8\right)} {\left(s + 5\right)}} \right\}$

$\frac{5}{21} \, e^{\left(-5 \, t\right)} - \frac{5}{21} \, e^{\left(-8 \, t\right)}$

#### Example 5 π

Explain how to use convolution along with the formula $$\mathcal{L}\{e^{at})=\frac{1}{s-a}$$to find the following inverse Laplace transform:

$\mathcal{L}^{-1}\left\{ \frac{3}{5 \, {\left(s + 2\right)} {\left(s - 7\right)}} \right\}$

$\frac{1}{15} \, e^{\left(7 \, t\right)} - \frac{1}{15} \, e^{\left(-2 \, t\right)}$

#### Example 6 π

Explain how to use convolution along with the formula $$\mathcal{L}\{e^{at})=\frac{1}{s-a}$$to find the following inverse Laplace transform:

$\mathcal{L}^{-1}\left\{ \frac{2}{3 \, {\left(s + 2\right)} {\left(s - 9\right)}} \right\}$

$\frac{2}{33} \, e^{\left(9 \, t\right)} - \frac{2}{33} \, e^{\left(-2 \, t\right)}$

#### Example 7 π

Explain how to use convolution along with the formula $$\mathcal{L}\{e^{at})=\frac{1}{s-a}$$to find the following inverse Laplace transform:

$\mathcal{L}^{-1}\left\{ \frac{5}{3 \, {\left(s - 2\right)} {\left(s - 6\right)}} \right\}$

$\frac{5}{12} \, e^{\left(6 \, t\right)} - \frac{5}{12} \, e^{\left(2 \, t\right)}$

#### Example 8 π

Explain how to use convolution along with the formula $$\mathcal{L}\{e^{at})=\frac{1}{s-a}$$to find the following inverse Laplace transform:

$\mathcal{L}^{-1}\left\{ \frac{5}{7 \, {\left(s + 7\right)} {\left(s - 2\right)}} \right\}$

$\frac{5}{63} \, e^{\left(2 \, t\right)} - \frac{5}{63} \, e^{\left(-7 \, t\right)}$

#### Example 9 π

Explain how to use convolution along with the formula $$\mathcal{L}\{e^{at})=\frac{1}{s-a}$$to find the following inverse Laplace transform:

$\mathcal{L}^{-1}\left\{ \frac{3}{5 \, {\left(s - 2\right)} {\left(s - 8\right)}} \right\}$

$\frac{1}{10} \, e^{\left(8 \, t\right)} - \frac{1}{10} \, e^{\left(2 \, t\right)}$

#### Example 10 π

Explain how to use convolution along with the formula $$\mathcal{L}\{e^{at})=\frac{1}{s-a}$$to find the following inverse Laplace transform:

$\mathcal{L}^{-1}\left\{ \frac{3}{2 \, {\left(s + 7\right)} {\left(s + 5\right)}} \right\}$

$\frac{3}{4} \, e^{\left(-5 \, t\right)} - \frac{3}{4} \, e^{\left(-7 \, t\right)}$

#### Example 11 π

Explain how to use convolution along with the formula $$\mathcal{L}\{e^{at})=\frac{1}{s-a}$$to find the following inverse Laplace transform:

$\mathcal{L}^{-1}\left\{ \frac{3}{5 \, {\left(s + 3\right)} {\left(s - 7\right)}} \right\}$

$\frac{3}{50} \, e^{\left(7 \, t\right)} - \frac{3}{50} \, e^{\left(-3 \, t\right)}$

#### Example 12 π

Explain how to use convolution along with the formula $$\mathcal{L}\{e^{at})=\frac{1}{s-a}$$to find the following inverse Laplace transform:

$\mathcal{L}^{-1}\left\{ \frac{3}{5 \, {\left(s + 6\right)} {\left(s + 5\right)}} \right\}$

$\frac{3}{5} \, e^{\left(-5 \, t\right)} - \frac{3}{5} \, e^{\left(-6 \, t\right)}$

#### Example 13 π

Explain how to use convolution along with the formula $$\mathcal{L}\{e^{at})=\frac{1}{s-a}$$to find the following inverse Laplace transform:

$\mathcal{L}^{-1}\left\{ \frac{7}{2 \, {\left(s - 2\right)} {\left(s - 7\right)}} \right\}$

$\frac{7}{10} \, e^{\left(7 \, t\right)} - \frac{7}{10} \, e^{\left(2 \, t\right)}$

#### Example 14 π

Explain how to use convolution along with the formula $$\mathcal{L}\{e^{at})=\frac{1}{s-a}$$to find the following inverse Laplace transform:

$\mathcal{L}^{-1}\left\{ \frac{3}{2 \, {\left(s - 4\right)} {\left(s - 7\right)}} \right\}$

$\frac{1}{2} \, e^{\left(7 \, t\right)} - \frac{1}{2} \, e^{\left(4 \, t\right)}$

#### Example 15 π

Explain how to use convolution along with the formula $$\mathcal{L}\{e^{at})=\frac{1}{s-a}$$to find the following inverse Laplace transform:

$\mathcal{L}^{-1}\left\{ \frac{2}{5 \, {\left(s + 5\right)} {\left(s - 7\right)}} \right\}$

$\frac{1}{30} \, e^{\left(7 \, t\right)} - \frac{1}{30} \, e^{\left(-5 \, t\right)}$

#### Example 16 π

Explain how to use convolution along with the formula $$\mathcal{L}\{e^{at})=\frac{1}{s-a}$$to find the following inverse Laplace transform:

$\mathcal{L}^{-1}\left\{ \frac{7}{5 \, {\left(s + 9\right)} {\left(s - 5\right)}} \right\}$

$\frac{1}{10} \, e^{\left(5 \, t\right)} - \frac{1}{10} \, e^{\left(-9 \, t\right)}$

#### Example 17 π

Explain how to use convolution along with the formula $$\mathcal{L}\{e^{at})=\frac{1}{s-a}$$to find the following inverse Laplace transform:

$\mathcal{L}^{-1}\left\{ \frac{7}{2 \, {\left(s - 1\right)} {\left(s - 8\right)}} \right\}$

$\frac{1}{2} \, e^{\left(8 \, t\right)} - \frac{1}{2} \, e^{t}$

#### Example 18 π

Explain how to use convolution along with the formula $$\mathcal{L}\{e^{at})=\frac{1}{s-a}$$to find the following inverse Laplace transform:

$\mathcal{L}^{-1}\left\{ \frac{3}{5 \, {\left(s - 2\right)} {\left(s - 6\right)}} \right\}$

$\frac{3}{20} \, e^{\left(6 \, t\right)} - \frac{3}{20} \, e^{\left(2 \, t\right)}$

#### Example 19 π

Explain how to use convolution along with the formula $$\mathcal{L}\{e^{at})=\frac{1}{s-a}$$to find the following inverse Laplace transform:

$\mathcal{L}^{-1}\left\{ \frac{5}{2 \, {\left(s + 6\right)} {\left(s - 1\right)}} \right\}$

$-\frac{5}{14} \, e^{\left(-6 \, t\right)} + \frac{5}{14} \, e^{t}$

#### Example 20 π

Explain how to use convolution along with the formula $$\mathcal{L}\{e^{at})=\frac{1}{s-a}$$to find the following inverse Laplace transform:

$\mathcal{L}^{-1}\left\{ \frac{5}{7 \, {\left(s + 7\right)} {\left(s + 1\right)}} \right\}$

$\frac{5}{42} \, e^{\left(-t\right)} - \frac{5}{42} \, e^{\left(-7 \, t\right)}$

#### Example 21 π

Explain how to use convolution along with the formula $$\mathcal{L}\{e^{at})=\frac{1}{s-a}$$to find the following inverse Laplace transform:

$\mathcal{L}^{-1}\left\{ \frac{2}{5 \, {\left(s + 8\right)} {\left(s + 3\right)}} \right\}$

$\frac{2}{25} \, e^{\left(-3 \, t\right)} - \frac{2}{25} \, e^{\left(-8 \, t\right)}$

#### Example 22 π

Explain how to use convolution along with the formula $$\mathcal{L}\{e^{at})=\frac{1}{s-a}$$to find the following inverse Laplace transform:

$\mathcal{L}^{-1}\left\{ \frac{2}{5 \, {\left(s + 7\right)} {\left(s + 2\right)}} \right\}$

$\frac{2}{25} \, e^{\left(-2 \, t\right)} - \frac{2}{25} \, e^{\left(-7 \, t\right)}$

#### Example 23 π

Explain how to use convolution along with the formula $$\mathcal{L}\{e^{at})=\frac{1}{s-a}$$to find the following inverse Laplace transform:

$\mathcal{L}^{-1}\left\{ \frac{2}{3 \, {\left(s + 9\right)} {\left(s - 3\right)}} \right\}$

$\frac{1}{18} \, e^{\left(3 \, t\right)} - \frac{1}{18} \, e^{\left(-9 \, t\right)}$

#### Example 24 π

Explain how to use convolution along with the formula $$\mathcal{L}\{e^{at})=\frac{1}{s-a}$$to find the following inverse Laplace transform:

$\mathcal{L}^{-1}\left\{ \frac{2}{7 \, {\left(s + 4\right)} {\left(s - 8\right)}} \right\}$

$\frac{1}{42} \, e^{\left(8 \, t\right)} - \frac{1}{42} \, e^{\left(-4 \, t\right)}$

#### Example 25 π

Explain how to use convolution along with the formula $$\mathcal{L}\{e^{at})=\frac{1}{s-a}$$to find the following inverse Laplace transform:

$\mathcal{L}^{-1}\left\{ \frac{2}{7 \, {\left(s - 3\right)} {\left(s - 8\right)}} \right\}$

$\frac{2}{35} \, e^{\left(8 \, t\right)} - \frac{2}{35} \, e^{\left(3 \, t\right)}$

#### Example 26 π

Explain how to use convolution along with the formula $$\mathcal{L}\{e^{at})=\frac{1}{s-a}$$to find the following inverse Laplace transform:

$\mathcal{L}^{-1}\left\{ \frac{7}{5 \, {\left(s + 6\right)} {\left(s - 2\right)}} \right\}$

$\frac{7}{40} \, e^{\left(2 \, t\right)} - \frac{7}{40} \, e^{\left(-6 \, t\right)}$

#### Example 27 π

Explain how to use convolution along with the formula $$\mathcal{L}\{e^{at})=\frac{1}{s-a}$$to find the following inverse Laplace transform:

$\mathcal{L}^{-1}\left\{ \frac{3}{5 \, {\left(s + 4\right)} {\left(s - 7\right)}} \right\}$

$\frac{3}{55} \, e^{\left(7 \, t\right)} - \frac{3}{55} \, e^{\left(-4 \, t\right)}$

#### Example 28 π

Explain how to use convolution along with the formula $$\mathcal{L}\{e^{at})=\frac{1}{s-a}$$to find the following inverse Laplace transform:

$\mathcal{L}^{-1}\left\{ \frac{2}{3 \, {\left(s + 5\right)} {\left(s - 7\right)}} \right\}$

$\frac{1}{18} \, e^{\left(7 \, t\right)} - \frac{1}{18} \, e^{\left(-5 \, t\right)}$

#### Example 29 π

Explain how to use convolution along with the formula $$\mathcal{L}\{e^{at})=\frac{1}{s-a}$$to find the following inverse Laplace transform:

$\mathcal{L}^{-1}\left\{ \frac{5}{7 \, {\left(s - 2\right)} {\left(s - 6\right)}} \right\}$

$\frac{5}{28} \, e^{\left(6 \, t\right)} - \frac{5}{28} \, e^{\left(2 \, t\right)}$

#### Example 30 π

Explain how to use convolution along with the formula $$\mathcal{L}\{e^{at})=\frac{1}{s-a}$$to find the following inverse Laplace transform:

$\mathcal{L}^{-1}\left\{ \frac{3}{7 \, {\left(s + 5\right)} {\left(s - 8\right)}} \right\}$

$\frac{3}{91} \, e^{\left(8 \, t\right)} - \frac{3}{91} \, e^{\left(-5 \, t\right)}$

#### Example 31 π

Explain how to use convolution along with the formula $$\mathcal{L}\{e^{at})=\frac{1}{s-a}$$to find the following inverse Laplace transform:

$\mathcal{L}^{-1}\left\{ \frac{3}{2 \, {\left(s + 9\right)} {\left(s + 4\right)}} \right\}$

$\frac{3}{10} \, e^{\left(-4 \, t\right)} - \frac{3}{10} \, e^{\left(-9 \, t\right)}$

#### Example 32 π

Explain how to use convolution along with the formula $$\mathcal{L}\{e^{at})=\frac{1}{s-a}$$to find the following inverse Laplace transform:

$\mathcal{L}^{-1}\left\{ \frac{3}{2 \, {\left(s + 1\right)} {\left(s - 7\right)}} \right\}$

$\frac{3}{16} \, e^{\left(7 \, t\right)} - \frac{3}{16} \, e^{\left(-t\right)}$

#### Example 33 π

Explain how to use convolution along with the formula $$\mathcal{L}\{e^{at})=\frac{1}{s-a}$$to find the following inverse Laplace transform:

$\mathcal{L}^{-1}\left\{ \frac{7}{2 \, {\left(s - 3\right)} {\left(s - 7\right)}} \right\}$

$\frac{7}{8} \, e^{\left(7 \, t\right)} - \frac{7}{8} \, e^{\left(3 \, t\right)}$

#### Example 34 π

Explain how to use convolution along with the formula $$\mathcal{L}\{e^{at})=\frac{1}{s-a}$$to find the following inverse Laplace transform:

$\mathcal{L}^{-1}\left\{ \frac{2}{3 \, {\left(s + 6\right)} {\left(s + 2\right)}} \right\}$

$\frac{1}{6} \, e^{\left(-2 \, t\right)} - \frac{1}{6} \, e^{\left(-6 \, t\right)}$

#### Example 35 π

Explain how to use convolution along with the formula $$\mathcal{L}\{e^{at})=\frac{1}{s-a}$$to find the following inverse Laplace transform:

$\mathcal{L}^{-1}\left\{ \frac{7}{2 \, {\left(s + 7\right)} {\left(s - 5\right)}} \right\}$

$\frac{7}{24} \, e^{\left(5 \, t\right)} - \frac{7}{24} \, e^{\left(-7 \, t\right)}$

#### Example 36 π

Explain how to use convolution along with the formula $$\mathcal{L}\{e^{at})=\frac{1}{s-a}$$to find the following inverse Laplace transform:

$\mathcal{L}^{-1}\left\{ \frac{2}{5 \, {\left(s + 7\right)} {\left(s - 5\right)}} \right\}$

$\frac{1}{30} \, e^{\left(5 \, t\right)} - \frac{1}{30} \, e^{\left(-7 \, t\right)}$

#### Example 37 π

Explain how to use convolution along with the formula $$\mathcal{L}\{e^{at})=\frac{1}{s-a}$$to find the following inverse Laplace transform:

$\mathcal{L}^{-1}\left\{ \frac{5}{7 \, {\left(s + 6\right)} {\left(s - 3\right)}} \right\}$

$\frac{5}{63} \, e^{\left(3 \, t\right)} - \frac{5}{63} \, e^{\left(-6 \, t\right)}$

#### Example 38 π

Explain how to use convolution along with the formula $$\mathcal{L}\{e^{at})=\frac{1}{s-a}$$to find the following inverse Laplace transform:

$\mathcal{L}^{-1}\left\{ \frac{2}{7 \, {\left(s + 7\right)} {\left(s - 5\right)}} \right\}$

$\frac{1}{42} \, e^{\left(5 \, t\right)} - \frac{1}{42} \, e^{\left(-7 \, t\right)}$

#### Example 39 π

Explain how to use convolution along with the formula $$\mathcal{L}\{e^{at})=\frac{1}{s-a}$$to find the following inverse Laplace transform:

$\mathcal{L}^{-1}\left\{ \frac{2}{3 \, {\left(s + 7\right)} {\left(s + 4\right)}} \right\}$

$\frac{2}{9} \, e^{\left(-4 \, t\right)} - \frac{2}{9} \, e^{\left(-7 \, t\right)}$

#### Example 40 π

Explain how to use convolution along with the formula $$\mathcal{L}\{e^{at})=\frac{1}{s-a}$$to find the following inverse Laplace transform:

$\mathcal{L}^{-1}\left\{ \frac{3}{2 \, {\left(s + 6\right)} {\left(s - 1\right)}} \right\}$

$-\frac{3}{14} \, e^{\left(-6 \, t\right)} + \frac{3}{14} \, e^{t}$

#### Example 41 π

Explain how to use convolution along with the formula $$\mathcal{L}\{e^{at})=\frac{1}{s-a}$$to find the following inverse Laplace transform:

$\mathcal{L}^{-1}\left\{ \frac{2}{5 \, {\left(s + 7\right)} {\left(s + 3\right)}} \right\}$

$\frac{1}{10} \, e^{\left(-3 \, t\right)} - \frac{1}{10} \, e^{\left(-7 \, t\right)}$

#### Example 42 π

Explain how to use convolution along with the formula $$\mathcal{L}\{e^{at})=\frac{1}{s-a}$$to find the following inverse Laplace transform:

$\mathcal{L}^{-1}\left\{ \frac{7}{2 \, {\left(s + 1\right)} {\left(s - 6\right)}} \right\}$

$\frac{1}{2} \, e^{\left(6 \, t\right)} - \frac{1}{2} \, e^{\left(-t\right)}$

#### Example 43 π

Explain how to use convolution along with the formula $$\mathcal{L}\{e^{at})=\frac{1}{s-a}$$to find the following inverse Laplace transform:

$\mathcal{L}^{-1}\left\{ \frac{2}{3 \, {\left(s - 3\right)} {\left(s - 9\right)}} \right\}$

$\frac{1}{9} \, e^{\left(9 \, t\right)} - \frac{1}{9} \, e^{\left(3 \, t\right)}$

#### Example 44 π

Explain how to use convolution along with the formula $$\mathcal{L}\{e^{at})=\frac{1}{s-a}$$to find the following inverse Laplace transform:

$\mathcal{L}^{-1}\left\{ \frac{3}{7 \, {\left(s + 7\right)} {\left(s + 5\right)}} \right\}$

$\frac{3}{14} \, e^{\left(-5 \, t\right)} - \frac{3}{14} \, e^{\left(-7 \, t\right)}$

#### Example 45 π

Explain how to use convolution along with the formula $$\mathcal{L}\{e^{at})=\frac{1}{s-a}$$to find the following inverse Laplace transform:

$\mathcal{L}^{-1}\left\{ \frac{2}{3 \, {\left(s + 5\right)} {\left(s - 9\right)}} \right\}$

$\frac{1}{21} \, e^{\left(9 \, t\right)} - \frac{1}{21} \, e^{\left(-5 \, t\right)}$

#### Example 46 π

Explain how to use convolution along with the formula $$\mathcal{L}\{e^{at})=\frac{1}{s-a}$$to find the following inverse Laplace transform:

$\mathcal{L}^{-1}\left\{ \frac{2}{3 \, {\left(s + 9\right)} {\left(s + 2\right)}} \right\}$

$\frac{2}{21} \, e^{\left(-2 \, t\right)} - \frac{2}{21} \, e^{\left(-9 \, t\right)}$

#### Example 47 π

Explain how to use convolution along with the formula $$\mathcal{L}\{e^{at})=\frac{1}{s-a}$$to find the following inverse Laplace transform:

$\mathcal{L}^{-1}\left\{ \frac{7}{3 \, {\left(s + 1\right)} {\left(s - 8\right)}} \right\}$

$\frac{7}{27} \, e^{\left(8 \, t\right)} - \frac{7}{27} \, e^{\left(-t\right)}$

#### Example 48 π

Explain how to use convolution along with the formula $$\mathcal{L}\{e^{at})=\frac{1}{s-a}$$to find the following inverse Laplace transform:

$\mathcal{L}^{-1}\left\{ \frac{2}{3 \, {\left(s + 5\right)} {\left(s - 7\right)}} \right\}$

$\frac{1}{18} \, e^{\left(7 \, t\right)} - \frac{1}{18} \, e^{\left(-5 \, t\right)}$

#### Example 49 π

Explain how to use convolution along with the formula $$\mathcal{L}\{e^{at})=\frac{1}{s-a}$$to find the following inverse Laplace transform:

$\mathcal{L}^{-1}\left\{ \frac{3}{7 \, {\left(s - 2\right)} {\left(s - 8\right)}} \right\}$

$\frac{1}{14} \, e^{\left(8 \, t\right)} - \frac{1}{14} \, e^{\left(2 \, t\right)}$
Explain how to use convolution along with the formula $$\mathcal{L}\{e^{at})=\frac{1}{s-a}$$to find the following inverse Laplace transform:
$\mathcal{L}^{-1}\left\{ \frac{7}{3 \, {\left(s + 8\right)} {\left(s - 2\right)}} \right\}$
$\frac{7}{30} \, e^{\left(2 \, t\right)} - \frac{7}{30} \, e^{\left(-8 \, t\right)}$