📚 Defining the Laplace Transform of Hyperbolic Functions
The Laplace Transform is a powerful tool for solving linear differential equations. It transforms a function of time, $f(t)$, into a function of a complex variable, $s$. When dealing with hyperbolic functions like $\sinh(at)$ and $\cosh(at)$, the Laplace Transform provides a straightforward way to handle these functions within differential equations.
📜 History and Background
The Laplace Transform is named after Pierre-Simon Laplace, who introduced it in his work on probability theory. It was later developed and used extensively by Oliver Heaviside in the context of solving differential equations arising in electrical engineering. The transform's ability to convert differential equations into algebraic equations made it an invaluable tool in various fields.
🔑 Key Principles
- 🔍Definition: The Laplace Transform of a function $f(t)$ is defined as:
$$F(s) = \mathcal{L}{f(t)} = \int_0^\infty e^{-st}f(t) dt$$
where $s$ is a complex number.
- 🔢Laplace Transform of $\sinh(at)$: Using the definition, we have:
$$\mathcal{L}{\sinh(at)} = \frac{a}{s^2 - a^2}, \quad s > |a|$$
This is derived from the exponential definition of $\sinh(at) = \frac{e^{at} - e^{-at}}{2}$.
- ➕Laplace Transform of $\cosh(at)$: Similarly,
$$\mathcal{L}{\cosh(at)} = \frac{s}{s^2 - a^2}, \quad s > |a|$$
This is derived from the exponential definition of $\cosh(at) = \frac{e^{at} + e^{-at}}{2}$.
- 💡Linearity: The Laplace Transform is a linear operator, meaning that for constants $c_1$ and $c_2$ and functions $f_1(t)$ and $f_2(t)$:
$$\mathcal{L}{c_1f_1(t) + c_2f_2(t)} = c_1\mathcal{L}{f_1(t)} + c_2\mathcal{L}{f_2(t)}$$
- 🔄Inverse Laplace Transform: If $F(s)$ is the Laplace Transform of $f(t)$, then the inverse Laplace Transform recovers $f(t)$ from $F(s)$, denoted as $f(t) = \mathcal{L}^{-1}{F(s)}$.
🌍 Real-world Examples
- 🌱Example 1: Consider the function $f(t) = 3\sinh(2t)$. Using the linearity property:
$$\mathcal{L}{3\sinh(2t)} = 3\mathcal{L}{\sinh(2t)} = 3\cdot \frac{2}{s^2 - 2^2} = \frac{6}{s^2 - 4}, \quad s > 2$$
- 🧪Example 2: Find the Laplace Transform of $g(t) = 2\cosh(3t) - \sinh(t)$.
$$\mathcal{L}{2\cosh(3t) - \sinh(t)} = 2\mathcal{L}{\cosh(3t)} - \mathcal{L}{\sinh(t)} = 2\cdot \frac{s}{s^2 - 9} - \frac{1}{s^2 - 1}, \quad s > 3$$
- 🔩Example 3: Solve the differential equation $y''(t) - y(t) = \cosh(t)$, with initial conditions $y(0) = 0$ and $y'(0) = 1$.
Applying the Laplace Transform:
$$s^2Y(s) - sy(0) - y'(0) - Y(s) = \frac{s}{s^2 - 1}$$
$$s^2Y(s) - 0 - 1 - Y(s) = \frac{s}{s^2 - 1}$$
$$(s^2 - 1)Y(s) = 1 + \frac{s}{s^2 - 1}$$
$$Y(s) = \frac{1}{s^2 - 1} + \frac{s}{(s^2 - 1)^2}$$
Taking the inverse Laplace Transform, we find $y(t)$.
📝 Conclusion
The Laplace Transform provides an efficient method for solving differential equations involving hyperbolic functions. By understanding the basic definitions, properties, and applying them to real-world examples, one can effectively utilize this tool in various engineering and mathematical problems. The linearity and transform rules for $\sinh(at)$ and $\cosh(at)$ are particularly useful in simplifying complex equations.