91Ó°ÊÓ

The general solution of a differential equation is a representation of all possible solutions that a differential equation can have. It embodies a family of curves or functions with one or more arbitrary constants, typically denoted by a symbol like \( C \). For the given differential equation \( \frac{d y}{d x} = \frac{x}{y} \), the process to find the general solution involves separating the variables, as seen in the first step of the solution.
After separation, we combine all terms involving \( y \) on one side and \( x \) on the other, yielding \( y \, dy = x \, dx \).
By integrating both sides, we are able to obtain the expression \( \frac{y^2}{2} = \frac{x^2}{2} + C \). This is our general solution before we apply any specific conditions.

• Integration includes finding antiderivatives, which introduces the integration constant \( C \).
• Each value of \( C \) represents a different curve from the family of solutions.

By multiplying both sides by 2, we further simplify the equation to \( y^2 = x^2 + 2C \). This equation now represents the general solution for the differential equation.

A particular solution of a differential equation results by substituting a specific condition into the general solution. It represents a single, unique solution curve, among the family of solutions described by the general solution.
In this exercise, the particular solution is based on the initial condition provided, which is \( y = 1 \) when \( x = 1 \).

• Substitute these values into the general solution \( y^2 = x^2 + 2C \) to find the specific \( C \).
• Doing this results in \( 0 = 2C \), leading to \( C = 0 \).
• Substituting \( C = 0 \) back into the general solution gives \( y^2 = x^2 \), which simplifies to two potential solutions, \( y = x \) or \( y = -x \).

Given the initial condition \( y = 1 \) when \( x = 1 \), the suitable particular solution is \( y = x \). It satisfies the initial condition and represents a specific path from our family of solutions.

Separation of variables is a fundamental technique for solving differential equations. It involves rearranging the equation so that all terms involving one variable (say \( y \)) are on one side while terms involving the other variable (\( x \)) are on the opposite side. This sets the stage for straightforward integration.
In our exercise, the original differential equation was \( \frac{d y}{d x} = \frac{x}{y} \). The separation technique led to \( y \, dy = x \, dx \).

• Each side of the equation only contains one variable, allowing independent integration.
• This method simplifies the integration process, making it easier to solve the equation analytically.

After separation, integrating each side gives us integral forms that lead smoothly toward the general solution. This step is critical for transforming complex differential equations into more manageable expressions.

An initial condition in a differential equation problem is a specified value for the function and often its derivatives at a particular point. It is essential to find a particular solution from the general solution.
In this exercise, the initial condition given is \( y = 1 \) when \( x = 1 \). When we plug these values into the general solution equation \( y^2 = x^2 + 2C \), it immediately helps determine the constant \( C \).

• Substituting the initial values gives \( 1^2 = 1^2 + 2C \), simplifying to \( 0 = 2C \), and hence \( C = 0 \).
• An initial condition ensures that the solution corresponds to the conditions in the actual problem context.
• Solving with this specific condition refines the general solution to describe a unique, specific curve, which in this case leads us to \( y = x \).

These conditions give a real-world context or a specific setting to the mathematical solution, honing down from infinite possibilities to a singular viable path.