91Ó°ÊÓ

A piecewise function is a function defined by multiple sub-functions, each applicable to a certain interval in the domain. In our differential equation, the function \( f(x) \) is defined piecewise with two branches:

• \( f(x) = x \) for \( 0 \leq x < 1 \)
• \( f(x) = 0 \) for \( x \geq 1 \)

This means that the behavior of the differential equation changes at \( x = 1 \), where \( f(x) \) shifts from a linear function to zero. Understanding how the piecewise components affect the behavior of the solution is crucial.When tackling a problem involving a piecewise function, it's important to address each segment independently, while ensuring solutions match at transition points. For our exercise, solutions for each segment are calculated separately and then adjusted to match at \( x = 1 \), maintaining continuity.

The integrating factor is a key concept when solving linear first-order differential equations. It simplifies the process of finding solutions by turning the equation into a form that can be easily integrated. The standard form of a linear first-order differential equation is \( \frac{dy}{dx} + P(x) y = Q(x) \).To solve such equations, we multiply through by an integrating factor \( \mu(x) = e^{\int P(x) \, dx} \), transforming the equation:
\[\mu(x) \frac{dy}{dx} + \mu(x)P(x) y = \mu(x)Q(x)\]
Here, the left-hand side becomes the derivative of \( \mu(x)y \). This makes the equation easy to integrate with respect to \( x \). In our case, for \( 0 \leq x < 1 \), the integrating factor is \( \mu(x) = e^{x^2} \). It turns our differential equation into a solvable form, leading to a solution valid for \( 0 \leq x < 1 \). By applying the integrating factor, we ensure that the solution process is streamlined and efficient.

Separable differential equations are a specific type of differential equation where the variables can be separated on each side of the equation. Such equations can be written in the form \( g(y) \, dy = h(x) \, dx \) and solved by integrating each side independently.In the step-by-step solution for \( x \geq 1 \), we encounter a differential equation \( \frac{dy}{dx} + 2xy = 0 \). This can be rearranged to

• \( \frac{dy}{y} = -2x \, dx \)

Both sides of the equation are then integrated separately, leading to logarithmic and polynomial expressions that can be combined to find a general solution.
The power of the separable differential equation method is its simplicity, allowing us to directly isolate and integrate the variables, resulting in straightforward solutions under appropriate conditions.

Ensuring the continuous solution across different segments is crucial when dealing with piecewise-defined functions. The objective is to make sure the solution behaves smoothly, with no jumps or gaps at critical points, such as \( x = 1 \) in our example.For our exercise, the solution was calculated separately for \( 0 \leq x < 1 \) and \( x \geq 1 \), and we brought them together at \( x = 1 \). We calculated the value of \( y \) at \( x = 1 \) from the \( 0 \leq x < 1 \) solution and used it to determine the constant \( k \) in the \( x \geq 1 \) solution. This ensured that the graph of the function is unbroken and smooth across the transition.Continuity is critical not just for mathematical rigor, but also for real-world problems where sudden changes in conditions could have significant implications. By ensuring continuous solutions, the behavior of dynamic systems can be accurately predicted and analyzed.