D3 - Inverse Laplace transforms


Example 1 πŸ”—

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

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

Answer:

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


Example 2 πŸ”—

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

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

Answer:

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


Example 3 πŸ”—

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

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

Answer:

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


Example 4 πŸ”—

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

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

Answer:

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


Example 5 πŸ”—

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

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

Answer:

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


Example 6 πŸ”—

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

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

Answer:

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


Example 7 πŸ”—

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

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

Answer:

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


Example 8 πŸ”—

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

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

Answer:

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


Example 9 πŸ”—

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

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

Answer:

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


Example 10 πŸ”—

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

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

Answer:

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


Example 11 πŸ”—

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

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

Answer:

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


Example 12 πŸ”—

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

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

Answer:

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


Example 13 πŸ”—

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

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

Answer:

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


Example 14 πŸ”—

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

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

Answer:

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


Example 15 πŸ”—

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

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

Answer:

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


Example 16 πŸ”—

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

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

Answer:

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


Example 17 πŸ”—

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

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

Answer:

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


Example 18 πŸ”—

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

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

Answer:

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


Example 19 πŸ”—

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

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

Answer:

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


Example 20 πŸ”—

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

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

Answer:

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


Example 21 πŸ”—

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

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

Answer:

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


Example 22 πŸ”—

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

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

Answer:

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


Example 23 πŸ”—

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

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

Answer:

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


Example 24 πŸ”—

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

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

Answer:

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


Example 25 πŸ”—

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

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

Answer:

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


Example 26 πŸ”—

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

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

Answer:

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


Example 27 πŸ”—

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

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

Answer:

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


Example 28 πŸ”—

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

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

Answer:

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


Example 29 πŸ”—

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

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

Answer:

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


Example 30 πŸ”—

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

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

Answer:

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


Example 31 πŸ”—

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

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

Answer:

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


Example 32 πŸ”—

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

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

Answer:

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


Example 33 πŸ”—

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

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

Answer:

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


Example 34 πŸ”—

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

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

Answer:

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


Example 35 πŸ”—

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

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

Answer:

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


Example 36 πŸ”—

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

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

Answer:

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


Example 37 πŸ”—

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

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

Answer:

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


Example 38 πŸ”—

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

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

Answer:

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


Example 39 πŸ”—

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

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

Answer:

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


Example 40 πŸ”—

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

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

Answer:

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


Example 41 πŸ”—

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

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

Answer:

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


Example 42 πŸ”—

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

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

Answer:

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


Example 43 πŸ”—

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

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

Answer:

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


Example 44 πŸ”—

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

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

Answer:

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


Example 45 πŸ”—

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

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

Answer:

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


Example 46 πŸ”—

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

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

Answer:

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


Example 47 πŸ”—

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

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

Answer:

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


Example 48 πŸ”—

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

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

Answer:

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


Example 49 πŸ”—

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

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

Answer:

\[ \frac{1}{14} \, e^{\left(8 \, t\right)} - \frac{1}{14} \, e^{\left(2 \, t\right)} \]


Example 50 πŸ”—

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

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

Answer:

\[ \frac{7}{30} \, e^{\left(2 \, t\right)} - \frac{7}{30} \, e^{\left(-8 \, t\right)} \]