91Ó°ÊÓ

Variable separation is a technique used to simplify the process of solving a differential equation. When you encounter a differential equation like \( \frac{\mathrm{d} y}{\mathrm{~d} x} = (x-y)^2 \), the aim is to separate the variables involved, resulting in a format where each side of the equation has only one variable: either \( x \) or \( y \).
This means you rearrange the terms such that all \( y \) terms, including \( \mathrm{d} y \), are on one side, and all \( x \) terms, including \( \mathrm{d} x \), are on the other. For this specific equation, you can achieve this by rewriting it as:

• \( \frac{\mathrm{d} y}{(x-y)^2} = \mathrm{d} x \).

This separation is crucial because it allows us to integrate each side independently, which is the next step in finding the solution to the differential equation.

Integration is the process of finding a function given its derivative. Once the differential equation is separated, the next step is to integrate both sides.
The left side is an integral with respect to \( y \), and the right side is an integral with respect to \( x \).

• The integral of \( \frac{1}{(x-y)^2} \) with respect to \( y \) gives us \( -\frac{1}{x-y} \).
• The integral of \( 1 \) with respect to \( x \) gives us \( x + C \), where \( C \) is the constant of integration.

Thus, integrating both sides gives us:
\[ -\frac{1}{x-y} = x + C \]
Integration is a powerful tool because it transforms a problem of finding a derivative into a problem of computing an anti-derivative, extending the method for solving equations expressed through their rates of change.

An initial condition is an extra piece of information given alongside a differential equation, allowing us to find the specific value of the constant of integration. In this context, the initial condition is given by \( y(1) = 1 \).
By substituting \( x = 1 \) and \( y = 1 \) into the integrated equation \( -\frac{1}{x-y} = x + C \), we attempt to solve for \( C \):

• \( -\frac{1}{1-1} = 1 + C \)

Unfortunately, this results in a mathematical problem, as division by zero is undefined, leading to complications and necessitating a re-evaluation of earlier steps or assumptions in the problem-solving sequence.

Initial conditions are essential as they enable us to take a general solution and transform it into a specific solution by solving for the constant.

In the process of integration, an arbitrary constant, known as the constant of integration, is introduced. When we integrate, say the function \( f(x) \), the result always includes a \( + C \), where \( C \) is this constant.
The constant of integration arises because the derivative of a constant is zero in the original function, hence it cannot be determined solely by the integration process and remains undetermined until an initial condition or boundary value is applied.
Re-evaluating based on the given initial condition, you solve \( -\frac{1}{x-y} = x + C \) for \( C \).

• Here, initial attempts to apply \( y(1)=1 \) led to issues primarily due to the setup leading to undefined terms.

Ultimately, through more careful analysis and recomputation, you find a viable solution \( C \), correcting initial missteps. The constant of integration ensures the complete family of solutions, and determines the specific solution applicable to a given initial condition or system state.