Solving Inverse Laplace Transforms of Irreducible Quadratic Factors using PFD.

📚 Understanding Inverse Laplace Transforms with Irreducible Quadratic Factors

The Inverse Laplace Transform is a powerful tool for solving differential equations, particularly those that arise in engineering and physics. When dealing with rational functions containing irreducible quadratic factors in the denominator, Partial Fraction Decomposition (PFD) becomes essential. An irreducible quadratic factor is one that cannot be factored further using real numbers.

📜 History and Background

The Laplace Transform, named after Pierre-Simon Laplace, has been used since the 18th century. Its inverse, however, gained prominence with the development of operational calculus by Oliver Heaviside. Partial Fraction Decomposition, a technique predating Laplace transforms, allows us to break down complex rational functions into simpler terms, making inversion easier.

🔑 Key Principles

• 🔍 Partial Fraction Decomposition (PFD): This involves expressing a rational function as a sum of simpler fractions. For an irreducible quadratic factor $(ax^2 + bx + c)$ in the denominator, the corresponding term in the PFD will be of the form $\frac{Ax + B}{ax^2 + bx + c}$.
• 📝 Irreducible Quadratic Factor: A quadratic expression $ax^2 + bx + c$ is irreducible if its discriminant, $b^2 - 4ac$, is negative. This means it has no real roots and cannot be factored using real numbers.
• 🧮 Completing the Square: Often, it's helpful to complete the square in the quadratic factor to simplify the inverse Laplace transform. That is, rewrite $ax^2 + bx + c$ as $a(x + h)^2 + k$, where $h = \frac{b}{2a}$ and $k = c - \frac{b^2}{4a}$.
• 💡 Standard Inverse Transforms: Familiarize yourself with standard inverse Laplace transforms, especially those involving sines and cosines, which arise from irreducible quadratic factors. For example, $\mathcal{L}^{-1} \left\{ \frac{a}{s^2 + a^2} \right\} = \sin(at)$ and $\mathcal{L}^{-1} \left\{ \frac{s}{s^2 + a^2} \right\} = \cos(at)$.

🧪 Real-World Examples

Let's work through a few examples to solidify the concept:

Example 1:

Find the inverse Laplace transform of $F(s) = \frac{3s + 5}{s^2 + 2s + 5}$.

Solution:

1. Complete the square: $s^2 + 2s + 5 = (s + 1)^2 + 4 = (s + 1)^2 + 2^2$.
2. Rewrite the numerator to match the form: $3s + 5 = 3(s + 1) + 2$.
3. Express $F(s)$ as: $F(s) = \frac{3(s + 1) + 2}{(s + 1)^2 + 2^2} = 3\frac{s + 1}{(s + 1)^2 + 2^2} + \frac{2}{(s + 1)^2 + 2^2}$.
4. Apply the inverse Laplace transform: $\mathcal{L}^{-1}{F(s)} = 3e^{-t}\cos(2t) + e^{-t}\sin(2t)$.

Example 2:

Find the inverse Laplace transform of $F(s) = \frac{2s - 1}{s^2 + 4}$.

Solution:

1. Rewrite the function: $F(s) = \frac{2s}{s^2 + 4} - \frac{1}{s^2 + 4}$.
2. Apply the inverse Laplace transform: $\mathcal{L}^{-1}{F(s)} = 2\cos(2t) - \frac{1}{2}\sin(2t)$.

Example 3:

Find the inverse Laplace transform of $F(s) = \frac{5}{s^2 + 6s + 13}$.

Solution:

1. Complete the square: $s^2 + 6s + 13 = (s + 3)^2 + 4 = (s + 3)^2 + 2^2$.
2. Rewrite the function: $F(s) = \frac{5}{(s + 3)^2 + 2^2} = \frac{5}{2} \cdot \frac{2}{(s + 3)^2 + 2^2}$.
3. Apply the inverse Laplace transform: $\mathcal{L}^{-1}{F(s)} = \frac{5}{2}e^{-3t}\sin(2t)$.

✍️ Practice Quiz

Try these practice problems to test your understanding:

1. Find the inverse Laplace transform of $\frac{s + 2}{s^2 + 2s + 2}$.
2. Find the inverse Laplace transform of $\frac{1}{s^2 + 4s + 5}$.
3. Find the inverse Laplace transform of $\frac{2s + 3}{s^2 + 6s + 10}$.
4. Find the inverse Laplace transform of $\frac{4}{s^2 + 9}$.
5. Find the inverse Laplace transform of $\frac{s}{s^2 + 16}$.
6. Find the inverse Laplace transform of $\frac{3s - 1}{s^2 + 8s + 20}$.
7. Find the inverse Laplace transform of $\frac{5}{s^2 + 25}$.

🎯 Conclusion

Solving inverse Laplace transforms with irreducible quadratic factors involves Partial Fraction Decomposition, completing the square, and recognizing standard inverse transforms. By practicing these techniques, you'll gain confidence in tackling complex problems in various fields, from electrical engineering to control systems. Keep practicing, and you'll master this essential skill!