(a) Find the equilibrium solution to the differential...

(a) Find the equilibrium solution to the differential equation \[ \frac{d y}{d t}=0.5 y-250 \] (b) Find the general solution to this differential equation. (c) Sketch the graphs of several solutions to this differential equation, using different initial values for $y$. (d) Is the equilibrium solution stable or unstable?

(a) Find the equilibrium solution to the differential equation
\[
\frac{d y}{d t}=0.5 y-250
\]
(b) Find the general solution to this differential equation.
(c) Sketch the graphs of several solutions to this differential equation, using different initial values for $y$.
(d) Is the equilibrium solution stable or unstable?

Key Concepts

Equilibrium Solution

The equilibrium solution of a differential equation is a constant solution where the derivative is zero. It represents a state in which the system, when undisturbed, remains steady without change over time. In analysis, finding the equilibrium solution involves setting the derivative equal to zero and solving for the dependent variable, thereby identifying a point in the phase space where the dynamics can settle.

General Solution to a First-Order Linear Differential Equation

The general solution to a first-order linear differential equation encompasses all possible solutions, typically including an arbitrary constant obtained through integration. Methods such as separation of variables or integrating factors are commonly used to derive the general solution. This solution describes the overall behavior of the system under various initial conditions.

Graphical Analysis of Differential Equation Solutions

Graphing solutions of a differential equation helps in visualizing the behavior of the system over time. By plotting different solution curves with varying initial conditions, one can observe trends, convergence to equilibrium, or divergence. This graphical representation provides insights into the dynamics of the system and how solutions evolve based on their starting values.

Stability Analysis of Equilibrium Solutions

Stability analysis involves determining whether solutions that start near an equilibrium state remain close to it over time or diverge away. An equilibrium is considered stable if small perturbations result in trajectories that stay near or return to the equilibrium, while it is unstable if slight deviations lead to significant divergence. This analysis is crucial in predicting the long-term behavior of dynamical systems.

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