To compute the inverse Laplace transform of the given expression:

$\mathcal{L}^{-1}\left(\frac{4}{p(5p + 1)}\right)$

Step-by-Step Solution:

1. Decompose the expression: The expression can be split using partial fraction decomposition. We aim to express the given fraction as a sum of simpler fractions.

   $\frac{4}{p(5p + 1)} = \frac{A}{p} + \frac{B}{5p + 1}$

   Multiply both sides by $p(5p + 1)$ to clear the denominator:

   $4 = A(5p + 1) + Bp$

   Now, expand the right-hand side:

   $4 = A \cdot 5p + A + Bp$

   Group terms by powers of $p$:

   $4 = (5A + B)p + A$

   To solve for $A$ and $B$, compare the coefficients of like powers of $p$:

   • For the $p^0$ term (constant term): $A = 4$
   • For the $p^1$ term: $5A + B = 0$, so $5 \times 4 + B = 0$, which gives $B = -20$

   Thus, the partial fraction decomposition is:

   $\frac{4}{p(5p + 1)} = \frac{4}{p} - \frac{20}{5p + 1}$

2. Find the inverse Laplace transform:

   We now compute the inverse Laplace transform of each term separately:

   • $\mathcal{L}^{-1}\left(\frac{4}{p}\right) = 4 \cdot 1 = 4$

   • $\mathcal{L}^{-1}\left(\frac{20}{5p + 1}\right)$: This matches the form $\frac{1}{p + a}$, whose inverse Laplace transform is $e^{-at}$. Here, $a = \frac{1}{5}$, so:

     $\mathcal{L}^{-1}\left(\frac{20}{5p + 1}\right) = 20e^{-\frac{t}{5}}$

3. Final result:

   Combine the two inverse Laplace transforms:

   $\mathcal{L}^{-1}\left(\frac{4}{p(5p + 1)}\right) = 4 - 20e^{-\frac{t}{5}}$

Final Answer:

$\mathcal{L}^{-1}\left(\frac{4}{p(5p + 1)}\right) = 4 - 20e^{-\frac{t}{5}}$

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Would you like more details or have any further questions?

5 Related Questions:

1. What is the inverse Laplace transform of $\frac{1}{p^2 + a^2}$?
2. How do you compute the Laplace transform of $t^n e^{at}$?
3. What are the properties of Laplace transforms used in solving differential equations?
4. Can you explain the convolution theorem in Laplace transforms?
5. How does partial fraction decomposition help in solving inverse Laplace transforms?

Tip:

When dealing with inverse Laplace transforms, partial fraction decomposition is a powerful technique to simplify complex fractions into forms with known transforms.