Question

Find the inverse transforms of the functions \(F(p)\). \(\frac{3 p+10}{p^{2}-25}\)

Short answer

The inverse Laplace transform is \[ \mathcal{L}^{-1} \left( F(p) \right) = 2.5 e^{5t} + 0.5 e^{-5t} \]

Step-by-step solution

Factor the Denominator

The given function is \[ F(p) = \frac{3p + 10}{p^2 - 25} \]First, factor the denominator. Notice that \[p^2 - 25 = (p - 5)(p + 5)\]So, the function becomes\[F(p) = \frac{3p + 10}{(p-5)(p+5)}\]

Decompose into Partial Fractions

Next, decompose the rational function into partial fractions. Assume\[ \frac{3p + 10}{(p-5)(p+5)} = \frac{A}{p-5} + \frac{B}{p+5} \]Multiply through by the denominator to clear the fractions:\[3p + 10 = A(p + 5) + B(p - 5)\]

Solve for Constants A and B

To find A and B, solve the system of equations by equating coefficients:Setting up the system:\[ 3p + 10 = A(p + 5) + B(p - 5) \]Choose convenient values for p. First, set p = 5:\[3(5) + 10 = A(5 + 5) + B(5 - 5) \Rightarrow 25 = 10A \Rightarrow A = 2.5 \]Then, set p = -5: \[3(-5) + 10 = A(-5 + 5) + B(-5 - 5) \Rightarrow -5 = -10B \Rightarrow B = 0.5\] Thus, the partial fractions are:\[ \frac{3p + 10}{(p-5)(p+5)} = \frac{2.5}{p-5} + \frac{0.5}{p+5} \]

Apply Inverse Laplace Transform

Now, take the inverse Laplace transforms of each term separately:\[ \mathcal{L}^{-1} \left( \frac{2.5}{p-5} \right) = 2.5 e^{5t} \]\[ \mathcal{L}^{-1} \left( \frac{0.5}{p+5} \right) = 0.5 e^{-5t} \]Add the results together:\[ \mathcal{L}^{-1} \left( F(p) \right) = 2.5 e^{5t} + 0.5 e^{-5t} \]

Key concepts

Partial Fraction Decomposition

Partial Fraction Decomposition is a method used to break down complex rational expressions into simpler fractions. This technique simplifies the process of finding the inverse Laplace Transform.
For instance, the function is given by \ F(p) = \frac{3p + 10}{p^2 - 25} \. To use Partial Fraction Decomposition, we first factor the denominator: $$p^2 - 25 = (p - 5)(p + 5)$$. Hence, we rewrite the function: $$F(p) = \frac{3p + 10}{(p-5)(p+5)}$$.
Now, we assume $$\frac{3p + 10}{(p-5)(p+5)} = \frac{A}{p-5} + \frac{B}{p+5}$$ and solve for constants A and B. This decomposition allows us to independently manage simpler fractions, making the Laplace Transform process much easier.

Laplace Transform Properties

Laplace Transform Properties are essential for efficiently working with transforms and their inverses. One must thoroughly understand these properties to apply the correct inverse transform.
Once the partial fractions are determined, employing Laplace Transform Properties helps in taking the inverse transform of each fraction.
For example, given the partial fractions $$\frac{2.5}{p-5}$$ and $$\frac{0.5}{p+5}$$, we use the inverse Laplace transform definitions:
- For $$\frac{A}{p-a}$$, the inverse Laplace is $$Ae^{at}$$.
Therefore:
$$ \mathcal{L}^{-1} \left( \frac{2.5}{p-5} \right) = 2.5 e^{5t} $$ and
$$ \mathcal{L}^{-1} \left( \frac{0.5}{p+5} \right) = 0.5 e^{-5t}$$.
Knowing these properties can significantly ease the process and improve understanding.

Solving Systems of Equations

Solving Systems of Equations is a fundamental algebraic method used to determine unknown constants in Partial Fraction Decomposition. In the given problem, we need to find constants A and B.
We start with the equation after assuming partial fractions: $$3p + 10 = A(p + 5) + B(p - 5)$$.
To solve for A and B efficiently, we substitute convenient values for p.
- First, set p = 5: $$3(5) + 10 = A(5 + 5)+ B(5 - 5)\Rightarrow A = 2.5 (as $$10A = 25$$)
- Then, set p = -5: $$3(-5) + 10 = A(-5 + 5) + B(-5 - 5) \Rightarrow B = 0.5 (as $$-10B = -5$$)
This method of substituting convenient values simplifies solving systems of equations, allowing us to find constants A and B to proceed with Partial Fraction Decomposition successfully.