Use the given general solution to find a solution of the...

Use the given general solution to find a solution of the differential equation having the given initial condition. Sketch the solution, the initial condition, and discuss the solution's interval of existence.
$$
y^{\prime}=y(2+y), y(t)=2 /\left(-1+C e^{-2 t}\right), y(0)=-3
$$

Key Concepts

Interval of Existence

The interval of existence refers to the range of values of the independent variable for which a given solution to a differential equation is valid and free of singularities. Determining this interval involves analyzing points where the solution might become undefined, such as where the denominator is zero, or where the right-hand side of the differential equation fails to satisfy the conditions required by the existence and uniqueness theorems.

Initial Value Problem (IVP)

An Initial Value Problem involves a differential equation along with a specified value of the unknown function at a particular point. This initial condition is used to determine the unique particular solution from the family of general solutions. It ensures that the solution not only satisfies the differential equation but also passes through a given point in the domain, thereby determining the constant of integration uniquely.

General vs. Particular Solutions

A general solution to a differential equation includes arbitrary constants that represent the infinite family of possible solutions. By applying an initial condition, one selectively chooses a particular solution that satisfies both the differential equation and the given initial condition. This process eliminates the arbitrary constants and provides a specific functional form for the solution.

Solution Sketching

Sketching the solution of a differential equation involves plotting the curve represented by the particular solution, marking the point corresponding to the initial condition, and indicating any asymptotic behavior or boundaries (due to the interval of existence). This visual representation aids in understanding the behavior of the solution over its domain and how it reacts near critical points or singularities.

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