Verify that the given function is a solution to the given...

Verify that the given function is a solution to the given differential equation $\left(c_{1} \text { and } c_{2}\right.$ are arbitrary constants), and state the maximum interval over which the solution is valid.
$$y(x)=\frac{1}{x+4}, \quad y^{\prime}=-y^{2}$$.

Key Concepts

Singularities

Singular points are values of the independent variable where the function or its derivative is not defined, often due to division by zero or other discontinuities. Identifying singularities helps in determining the maximum interval of validity for the solution, ensuring that the solution is only considered over intervals where it is well-behaved.

Interval of Validity

The interval of validity refers to the largest domain in which the solution to a differential equation is defined and continuously differentiable. It is crucial to identify this domain because the solution may not be valid at points where the function is undefined or where discontinuities occur.

Differential Equations

A differential equation involves unknown functions and their derivatives, and it is used to model phenomena where rates of change are related to the function itself. Understanding these equations and their structure is essential for determining if a given function fulfills the relationship imposed by the derivative rule specified in the equation.

Verification of Solutions

Verifying a solution to a differential equation involves substituting the proposed function into the equation to demonstrate that it satisfies the equation identically. This process confirms that the function’s derivative and the functional form match the requirements imposed by the differential equation.

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