Question

For the following problems, use a software program or your calculator to generate the directional fields. Solve explicitly and draw solution curves for several initial conditions. Are there some critical initial conditions that change the behavior of the solution?[I] \(y^{\prime}=e^{y}\)

Short answer

There are no critical initial conditions; solutions exponentially diverge for positive \(y\).

Step-by-step solution

Understand the Differential Equation

The given differential equation is \(y' = e^y\). This is a first-order differential equation that expresses the derivative \(y'\) as a function of \(y\). Our task is to analyze its behavior using directional fields and to solve it explicitly.

Generate the Directional Field

Using a calculator or software like Desmos, Python, or Mathematica, generate the directional field for \(y' = e^y\). This field will show small line segments representing the slope \(e^y\) at various points \((x, y)\) on the plane. Observing this, we can see how solutions potentially behave at different initial conditions.

Solve the Differential Equation Explicitly

To solve \(y' = e^y\) explicitly, separate the variables: \(\frac{dy}{e^y} = dx\). Now integrate both sides: \(\int rac{1}{e^y} \, dy = \int \, dx\). This gives \(-rac{1}{e^y} = x + C\), where \(C\) is the constant of integration. Solving for \(y\), we find \(y = -\ln(x + C)\).

Analyze Solution Curves

Plot the solution curves for several initial conditions. Solutions will take the form \(y = -\ln(x + C)\), where each different \(C\) corresponds to a different initial condition. The shape of these curves depends on the value of \(C\), and they should match the directional field.

Discuss Critical Initial Conditions

Determine if there are critical initial conditions that change the solution's behavior. In this case, there are no distinct critical points like equilibrium solutions since \(y' = e^y\) only has the behavior of increasing slopes as \(y\) becomes more positive. The function \(e^y\) is always positive, which causes solutions to diverge as \(y\) increases, showing that initial conditions primarily affect the growth rate.

Key concepts

Directional Fields

Directional fields, also referred to as slope fields, are visual representations that help us understand how differential equations behave. They consist of a multitude of small segments drawn on the coordinate plane. Each segment shows the slope of the solution at that particular point.
This graphical method is particularly helpful because:

• It reveals the general direction in which the solutions move.
• It allows us to predict the behavior of solutions without solving the equation explicitly.
• It gives insight into the stability and behavior over different sets of conditions through visual cues.

In our problem, the directional field for the equation y' = e^y provides an understanding of how solutions might flow. As you generate this field using software or a graphing calculator, you notice that the lines curve upwards, indicating that as the value of y increases, so does the slope of the solution.
This visually confirms the positivity of e^y, highlighting that all solutions will tend to rise steeply or diverge as they proceed.

Initial Conditions

Initial conditions are crucial because they determine the specific solution to the differential equation. Simply put, while a differential equation might have infinitely many solutions, an initial condition helps pin down exactly which path the solution will take.
In our example, even though the general integrated solution is y = -ln(x + C), where C is a constant, the specific path of each solution curve is determined by the initial conditions you apply.
These conditions are values given at the start, such as x = x₀ and y = y₀. By doing so, you essentially determine a particular solution (a particular value for C), which then dictates the behavior of the entire curve.

• For instance, starting with a pair of coordinates such as (0, 1) will dictate a unique curve that matches the initial condition exactly.
• This ensures that while the directional field shows all potential paths, initial conditions select exactly one of those paths.

Solution Curves

Solution curves are the graphical representations of solutions that match both the differential equation and any imposed initial conditions. These curves are derived from integrating the differential equation and choosing particular values based on initial conditions.
For our equation, the expression y = -ln(x + C) represents the family of solution curves. Each unique C corresponds to a different initial condition:

• By varying the constants, we get different curves, each passing through initial point (x₀, y₀).
• These curves help visualize the path of particles in a fluid dynamics setting or represent population growth under a given model.

When you overlay these solution curves onto a directional field plot, it becomes evident how each solution line tends to follow the direction of the small slope line segments, reinforcing the relationship between the graphical tips provided by the slope field and the explicit solutions.

Variable Separation

Variable separation is a traditional and effective method used to solve first-order differential equations like the one in our problem, y' = e^y.
The main idea here is to manipulate the equation in such a way that all terms involving y are on one side and all terms involving x are on the other side. This process is often referred to as 'separating the variables'.
Let's break it down:

• Rearrange the given equation as dy/e^y = dx. This separation makes it easier to integrate both sides independently.
• Integrate both sides: \(\int \frac{1}{e^y} \, dy = \int \, dx\).
• The integration yields a solution in terms of logarithms: -1/e^y = x + C, where C is the integration constant.
• Solving for y gives y = -ln(x + C), describing the complete set of solutions.

This technique not only highlights the structure of the equation but also simplifies the process of finding solutions, especially for equations where variable separation is applicable.