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The order of a differential equation is determined by the highest derivative present in the equation. It's important because it reveals the complexity and type of the equation we are dealing with. For example, in part (a), the differential equation is \(2 \frac{d y}{d x} + y = x - 1\). Here, the highest derivative present is the first derivative \(\frac{d y}{d x}\), making it a first-order differential equation. Meanwhile, in part (b), the differential equation \(y'' - y = 0\) involves the second derivative \(y''\). Therefore, it is classified as a second-order differential equation.Understanding the order is crucial for solving these equations because:

• It dictates the number of initial or boundary conditions required for a unique solution.
• Higher-order equations generally require more complex solution methods.

Each order has its typical characteristics and methods of solution. If you're comfortable identifying the order, you're already on your way to mastering differential equations.

A first derivative, often denoted as \(\frac{d y}{d x}\) or \(f'(x)\), signifies how a function \(y\) changes with respect to \(x\). It essentially provides the slope or rate of change of \(y\) at any given point. In the context of differential equations, such as \(2 \frac{d y}{d x} + y = x - 1\), the presence of the first derivative means examining linear systems involving one variable.Calculating the first derivative is fundamental because:

• It helps understanding how rapidly a function's values are changing.
• It forms the basis for finding extrema like maxima, minima, and points of inflection.

For instance, in verifying solutions for part (a), finding \( \frac{d y}{d x} = -\frac{1}{2} c e^{-x / 2} + 1 \) is a crucial step to see if the proposed solution fits the differential equation.

The second derivative, noted as \(y''\) or \(f''(x)\), measures the rate at which the first derivative changes. This offers insight into the curvature or concavity of the original function. It is especially pivotal in understanding acceleration in motion or changes in trends.Consider the differential equation \(y'' - y = 0\) from part (b). This focuses on how the change in the rate of change of our function \(y\) behaves. When calculating solutions, you find \(y'' = c_1 e^t + c_2 e^{-t}\), and comparing \(y''\) back to \(y\), helps verify that the given function is a solution by illustrating that both sides zero out.The significance of second derivatives is manifold:

• It determines the concavity or convexity of a graph.
• Plays a key role in physics and engineering for understanding systems' responses, like oscillations.

Grasping the concept of second derivatives allows us to not just solve equations but predict how systems will behave under different conditions.