38–43. Equilibrium solutions A differential equation of the form y′(t)=f(y) is said to be autonomous (the function f depends only on y). The constant function y=y0 is an equilibrium solution of the equation provided f(y0)=0 (because then y'(t)=0 and the solution remains constant for all t). Note that equilibrium solutions correspond to horizontal lines in the direction field. Note also that for autonomous equations, the direction field is independent of t. Carry out the following analysis on the given equations.
a. Find the equilibrium solutions.


y′(t) = 2y + 4

Verified step by step guidance

1. Identify the given differential equation: \(y'(t) = 2y + 4\).
2. Recall that equilibrium solutions occur when the derivative \(y'(t)\) is zero, meaning the function \(y(t)\) does not change over time.
3. Set the right-hand side of the differential equation equal to zero to find equilibrium points: \$2y + 4 = 0$.
4. Solve the algebraic equation for \(y\) to find the equilibrium solution(s).
5. Interpret the solution(s) as constant functions \(y(t) = y_0\) where \(y_0\) satisfies the equation, representing horizontal lines in the direction field.

Key Concepts

Autonomous Differential Equations

An autonomous differential equation is one where the derivative y' depends only on the variable y, not explicitly on the independent variable t. This means the rate of change of y is determined solely by y itself, simplifying analysis and allowing the use of phase line methods.

Equilibrium Solutions

Equilibrium solutions occur when y' = 0, meaning the function y(t) remains constant over time. For autonomous equations y' = f(y), equilibrium points are found by solving f(y) = 0. These solutions correspond to horizontal lines in the direction field where the system is at rest.

Direction Fields and Stability

Direction fields graphically represent the slope y' at various points (t, y). For autonomous equations, slopes depend only on y, producing horizontal patterns independent of t. Equilibrium solutions appear as horizontal lines, and analyzing nearby slopes helps determine their stability.