By using Laplace transforms, solve the following differential equations subject to the given initial conditions. $$y^{\prime \prime}-8 y^{\prime}+16 y=32 t, \quad y_{0}=1, y_{0}^{\prime}=2$$

Differential equations involve functions and their derivatives. These equations showcase how a function changes with respect to an independent variable, typically time. For example, in physics, they describe how velocity changes over time, given acceleration. Differential equations are categorized based on their order (first, second, etc.) and linearity. The given exercise deals with a second-order linear differential equation: \(y'' - 8y' + 16y = 32t\). Here, we have the second derivative \(y''\), the first derivative \(y'\), and the function itself \(y\). Solving differential equations is crucial for modeling phenomena across various fields like engineering, physics, and economics.

Initial conditions specify the values of the function and its derivatives at the start. These are essential in finding unique solutions. In our exercise, the initial conditions are \(y(0) = 1\) and \(y'(0) = 2\). They help in simplifying the transformed differential equation. When we take the Laplace transform of the second-order differential equation, we use these initial conditions for solving the resulting algebraic equation. Initial conditions often depict starting points, such as initial temperature, population, or velocity, making the solution uniquely determined.

The Inverse Laplace Transform is used to revert back from the frequency domain to the time domain. The process involves rules and tables for common transforms. In our exercise, after solving the algebraic equation for \(Y(s)\) in the Laplace domain, we need the inverse Laplace transform to find \(y(t)\). For instance, \(\mathcal{L}^{-1} \left\{ \frac{8}{s^2} \right\} = 8t\) and \(\mathcal{L}^{-1} \left\{ \frac{8}{(s - 4)^2} \right\} = 8te^{4t}\). Understanding these inverses is crucial since it brings the physical interpretation back from the mathematical abstraction.