Proceed as in Problems 29-32 to determine whether the given...

Proceed as in Problems $29-32$ to determine whether the given differential equation possesses constant solutions. Solve the given differential equation and then find a solution whose graph passes through the indicated point. $x \frac{d y}{d x}=y^{2}-y$ (a) (0,1) (b) (0,0) (c) $\left(\frac{1}{2}, \frac{1}{2}\right)$

Proceed as in Problems $29-32$ to determine whether the given differential equation possesses constant solutions. Solve the given differential equation and then find a solution whose graph passes through the indicated point.
$x \frac{d y}{d x}=y^{2}-y$
(a) (0,1)
(b) (0,0)
(c) $\left(\frac{1}{2}, \frac{1}{2}\right)$

Key Concepts

Constant Solutions

This concept involves identifying constant functions that satisfy a differential equation by setting the derivative equal to zero. In the context of the problem, constant or equilibrium solutions occur when the expression on the right-hand side is zero, leading to values of the dependent variable that do not change with respect to the independent variable.

Separable Differential Equations

A separable differential equation is one in which the variables can be rearranged so that all terms containing one variable and its differential are on one side of the equation and all terms containing the other variable and its differential on the opposite side. This allows direct integration of both sides, which is a core method for solving such equations.

Partial Fractions

When integrating rational functions, particularly in separable equations where the integrand is a ratio of polynomials, partial fraction decomposition is used to break down the integrand into simpler fractions that are easier to integrate. This technique is essential when the denominator factors into linear or easily integrable quadratic expressions.

Initial Value Problems

Initial value problems involve finding a specific solution to a differential equation that satisfies a given condition at a particular point. After obtaining the general solution by integration, the constant of integration is determined by applying the initial condition, ensuring that the solution’s graph passes through the prescribed point.

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