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A general solution to a differential equation is a family of solutions that includes all possible specific solutions a differential equation can have. For the given differential equation \( p^2 - x p + y = 0 \), where \( p = \frac{dy}{dx} \), applying the quadratic formula is key to determining the general solution. The quadratic formula is given by:

• \( p = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)

Here, \( a = 1 \), \( b = -x \), and \( c = y \). This positions the differential equation as a quadratic one in terms of \( p \). The general solution is obtained when the discriminant \( x^2 - 4y > 0 \), which indicates there are two real and distinct solutions. These solutions are two functions that derive from \( p = \frac{x \pm \sqrt{x^2 - 4y}}{2} \). These functions represent a wide range of curves that satisfy the original differential equation for any value of the constants involved.

A singular solution of a differential equation is a specific solution that cannot be derived from the general solution by specifying particular values of arbitrary constants. To determine if a singular solution exists for the equation \( p^2 - x p + y = 0 \), we examine the discriminant \( x^2 - 4y \).

• If the discriminant is equal to zero \( x^2 - 4y = 0 \), the equation has a double root.
• This root corresponds to a repeated solution, \( p = \frac{x}{2} \).

This indicates the presence of a singular solution, as this solution does not belong to the two families formed when the discriminant is positive. Solving \( x^2 - 4y = 0 \) yields \( y = \frac{x^2}{4} \). This particular solution stands alone and provides a unique curve on the graph of solutions, distinct from the general solutions.

Understanding the role of a quadratic equation in finding solutions to differential equations is essential. A quadratic equation has the form \( ax^2 + bx + c = 0 \).

• In our problem, for \( p^2 - x p + y = 0 \), it represents a quadratic equation with respect to \( p \), the derivative \( \frac{dy}{dx} \).
• The coefficients are \( a = 1 \), \( b = -x \), and \( c = y \).
• The quadratic equation provides mechanisms for determining possible values of \( p \) through its solutions.

The two potential solutions of this quadratic equation represent possible forms of the derivative \( \frac{dy}{dx} \), leading to different expressions for the function \( y \). Essentially, solving the quadratic equation helps us to find and express the general and potential singular solutions to the differential question.

The discriminant is a crucial component derived from the quadratic equation that informs us about the nature of the equation's roots. It is calculated using the expression \( b^2 - 4ac \).

• For our equation \( p^2 - x p + y = 0 \), which is rewritten as a quadratic in \( p \), the discriminant is \( (-x)^2 - 4(1)(y) = x^2 - 4y \).
• A positive discriminant \( (x^2 - 4y > 0) \) signifies two real and distinct solutions.
• A zero discriminant \( (x^2 - 4y = 0) \) implies a repeated solution, indicating a potential singular solution.
• If the discriminant were negative, it would mean the absence of real solutions for \( p \).

The discriminant is essential as it dictates whether a singular solution might exist and the kind of general solutions the differential equation can provide. Thus, analyzing the discriminant is a key step in understanding the nature of solutions available for any given quadratic equation in differential equations.