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. $\frac{d y}{d x}=y^{2}-9$ (a) (0,0) (b) (0,3) (c) $\left(\frac{1}{3}, 1\right)$

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.
$\frac{d y}{d x}=y^{2}-9$
(a) (0,0)
(b) (0,3)
(c) $\left(\frac{1}{3}, 1\right)$

Key Concepts

Constant Solutions

Constant solutions (or equilibrium solutions) are those solutions of a differential equation where the dependent variable remains fixed at a certain value, meaning that its derivative is zero. To determine these in any differential equation, one sets the derivative equal to zero and solves for the dependent variable, identifying the values for which the system is in balance.

Separation of Variables

Separation of variables is a fundamental technique for solving differential equations in which the equation is rearranged so that all terms involving the dependent variable are on one side and all terms involving the independent variable are on the other side. Once separated, it becomes possible to integrate both sides independently to find the general solution.

Partial Fraction Decomposition

Partial fraction decomposition is a method used in calculus to simplify the integration of rational functions by breaking them into simpler, easier-to-integrate terms. This technique is particularly useful when the integration resulting from the separation of variables involves a rational function that can be expressed as a sum of simpler fractions.

Initial Value Problems

Initial value problems involve finding a specific solution to a differential equation that satisfies a given condition at a particular point, known as the initial condition. This allows one to identify a unique solution from a family of possible general solutions, ensuring the solution passes through a specified point.

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