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.
c. Sketch the solution curve that corresponds to the initial condition y0=1. 


y′(t) = 2y + 4

Verified step by step guidance

1. Identify the given differential equation: \(y'(t) = 2y + 4\). This is an autonomous equation because the right-hand side depends only on \(y\), not explicitly on \(t\).
2. Find the equilibrium solutions by setting \(y'(t) = 0\), which means solving \$2y + 4 = 0\( for \)y\(. This will give the constant values of \)y$ where the solution does not change over time.
3. Solve the equation \$2y + 4 = 0\( to find the equilibrium solution(s) \)y_0$. This equilibrium solution corresponds to a horizontal line in the direction field.
4. Since the initial condition is \(y(0) = 1\), analyze the behavior of the solution near \(y=1\) by considering the sign of \(y'(t)\) when \(y=1\). This will tell you whether the solution is increasing or decreasing at that point.
5. Sketch the solution curve starting at the point \((0,1)\) on the \(t\)-\(y\) plane. Use the information about the equilibrium solution and the slope \(y'(t)\) at \(y=1\) to draw the curve showing how \(y\) changes with \(t\).

Key Concepts

Autonomous Differential Equations

An autonomous differential equation has the form y' = f(y), where the rate of change depends only on y, not explicitly on t. This means the behavior of solutions depends solely on the current value of y, making the direction field time-invariant and simplifying analysis of solution curves.

Equilibrium Solutions

Equilibrium solutions occur when y' = f(y) = 0, meaning the solution y(t) remains constant over time. These correspond to horizontal lines in the direction field and represent steady states where the system does not change, providing key reference points for sketching solution behavior.

Sketching Solution Curves with Initial Conditions

To sketch a solution curve for a given initial condition y(0) = y0, identify equilibrium points and analyze the slope y' = f(y) at y0. The sign and magnitude of y' determine whether the solution increases or decreases, guiding the shape of the curve over time.