Question

A differential equation of the form \(y^{\prime}(t)=f(y)\) is said to be autonomous (the function \(f\) depends only on \(y\) ). The constant function \(y=y_{0}\) is an equilibrium solution of the equation provided \(f\left(y_{0}\right)=0\) (because then \(y^{\prime}(t)=0\) 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. b. Sketch the direction field, for \(t \geq 0\). c. Sketch the solution curve that corresponds to the initial condition \(y(0)=1\). $$y^{\prime}(t)=6-2 y$$

Short answer

Answer: The equilibrium solution of the given differential equation is \(y = 3\). The solution curve corresponding to the initial condition \(y(0) = 1\) will have an increasing slope below the equilibrium solution at \(y = 3\), and it will approach the equilibrium solution as \(t\) goes to infinity.

Step-by-step solution

a. Finding Equilibrium Solutions#End TITLE#

To find the equilibrium solutions, set the derivative equal to zero, i.e., \(y'(t) = 0\), and solve for \(y\): $$0 = 6 - 2y$$ Therefore, the equilibrium solution is: $$y = 3$$#End Content#

b. Sketching the Direction Field#End TITLE#

First, let's find the behavior for points \(y\) above and below the equilibrium solution \(y = 3\). For \(y < 3\), $$y' = 6 - 2y > 0 \Rightarrow y(t) \text{ is increasing for } y < 3$$ For \(y > 3\), $$y' = 6 - 2y < 0 \Rightarrow y(t) \text{ is decreasing for } y > 3$$ Now, sketch the direction field for \(t \geq 0\) with: - Horizontal lines at \(y=3\) (equilibrium solution). - Arrows pointing up below \(y=3\) and down above \(y=3\).#End Content#

c. Sketching the Solution Curve with Initial Condition \(y(0) = 1\)#End TITLE#

Integrating the given differential equation, $$\frac{dy}{y'(t)} = \frac{dy}{6-2y} = dt$$ Using partial fraction decomposition, $$\frac{dy}{6-2y} = \frac{1}{3} \left(\frac{2}{3-y}-\frac{2}{y}\right) dy = dt$$ Integrating both sides, $$\frac{1}{3}\left(-\frac{2}{3}\ln|3-y|+2\ln|y|\right) = t+C$$ Now, we apply the initial condition \(y(0) = 1\). Putting the value in the equation $$\frac{1}{3}\left(-\frac{2}{3}\ln|2|+2\ln|1|\right) = 0+C$$ So, \(C = \frac{2}{3}\ln{2}\). Replacing \(C\) back in the equation, we have, $$\frac{1}{3}\left(-\frac{2}{3}\ln|3-y|+2\ln|y|\right) = t+\frac{2}{3}\ln{2}$$ Now, plot the solution curve that corresponds to this equation using the initial condition \(y(0)=1\): - Draw the curve passing through \((0,1)\). - The curve should have an increasing slope below the equilibrium solution at \(y=3\). - The curve should approach the equilibrium solution \(y=3\) as \(t\) goes to infinity. #End Content#

Key concepts

Equilibrium Solutions

Understanding equilibrium solutions in autonomous differential equations is crucial. These solutions occur when the derivative of a function, given by \(y'(t)\), equals zero. When this happens, the function does not change over time and remains constant. For example, in the equation \(y'(t) = 6 - 2y\), the equilibrium solution is found by setting \(6 - 2y = 0\). Solving for \(y\), we find that \(y = 3\). Equilibrium solutions illustrate where the function levels off and can be represented as horizontal lines in a direction field. These lines show the points at which the rate of change of \(y\) is zero, indicating stability at that particular value of \(y\). Such solutions are fundamental for studying the behavior of differential equations as they may suggest long-term behavior of the system modeled by the equation.

Direction Field

A direction field is a powerful visual tool for analyzing differential equations. For autonomous equations like \(y'(t) = 6 - 2y\), the direction field is independent of \(t\), meaning the field's orientation relies solely on the \(y\)-values. The direction field helps visualize how the solutions to the differential equation behave over time.In this example, at \(y = 3\), the direction arrows are horizontal, denoting the equilibrium. Above this line (\(y > 3\)), the vectors point downward indicating that the solutions decrease over time. Below \(y = 3\) (\(y < 3\)), the vectors point upwards, hinting at an increase over time. With this information, one can sketch a confident prediction of how different initial conditions will evolve through time, giving a clear picture of the long-term trends of solution curves.

Initial Conditions

In differential equations, initial conditions dictate the starting point of the solution curve. Consider the initial condition \(y(0) = 1\) for the equation \(y'(t) = 6 - 2y\). This condition tells us where the function starts when \(t = 0\).Applying the initial condition means we need to find the specific curve that not only satisfies the differential equation but also starts from this given point. For \(y'(t) = 6 - 2y\), starting at \(y(0) = 1\) means plotting the solution curve so that it intersects the y-axis at 1.Initial conditions are integral in determining the unique trajectory that the solution curve will follow over time. They personalize the general behavior predicted by the equilibrium solution and the direction field to fit specific starting parameters.

Solution Curves

Solution curves represent the paths described by the solutions of a differential equation over time and are drawn by integrating the differential equation. With \(y'(t) = 6 - 2y\), we integrate to find the specific function of \(y(t)\) given initial conditions. For instance, starting from \(y(0) = 1\), we integrate to find the path that starts at this point. The solution curve will show how the value of \(y\) evolves as \(t\) increases. Solution curves provide a comprehensive view of how solutions transition toward or away from equilibrium solutions over time. They show whether the system stabilizes, grows, or decays, which is essential for understanding the dynamics of autonomous differential equations. By plotting these curves, we visualize the entire solution across a timeline, illustrating the behavior dictated by the initial conditions and the direction field.