Differential Equations - Sketching Phase Curves

Topic: Differential Equations - Sketching Phase Curves

Differential equations involve the relationship between a function and its derivatives.
Phase curves in differential equations help visualize the behavior of the functions.
Sketching phase curves help understand the solution and analyze its properties.
We will learn the steps to sketch phase curves in this lecture.

Step 1: Analyze the Differential Equation

Begin by analyzing the differential equation provided.
Identify the highest order of the derivative present.
Determine if the given equation is linear or nonlinear.
Classify the equation into categories such as first order, second order, etc.

Step 2: Find Critical Points

Locate any critical points or equilibrium points in the equation.
These points are obtained by setting the derivative(s) equal to zero.
Critical or equilibrium points help identify the different regions and behaviors of the solution.

Step 3: Plot Critical Points

Plot the critical or equilibrium points on the phase plane.
Use different symbols or colors to represent different types of critical points.
For example, stable points can be represented by filled circles, unstable points by open circles, and so on.

Step 4: Analyze the Behavior near Critical Points

Use linearization techniques to analyze the behavior near critical points.
Linearize the equation using partial derivatives and evaluate at each critical point.
Determine if the critical point is stable, unstable, or semi-stable.
This analysis helps understand the nature of the solution near critical points.

Step 5: Determine Asymptotic Behavior

Determine the asymptotic behavior of the solution.
Analyze whether the solution approaches any specific values or tends to infinity or zero in the long run.
This information helps in drawing the phase curves accurately.

Step 6: Draw Representative Solutions

Choose appropriate initial conditions to draw representative solutions.
Evaluate the differential equation with these initial conditions to obtain the solution.
Sketch the solution curves on the phase plane, considering the behaviors analyzed previously.

Step 7: Analyze Stability

Analyze the stability of the solutions based on their behavior and the critical points.
Stable solutions approach a critical point, while unstable solutions diverge from it.
Semi-stable solutions exhibit a combination of stability and instability.
This analysis helps understand the overall behavior of the system.

Example: Sketching Phase Curves

Consider a first-order linear differential equation: dy/dx = -2y + 4.
Analyze the equation and determine its critical points and behavior.
Plot the critical points on the phase plane and analyze stability.
Draw representative solutions and analyze their stability based on the behaviors observed.

Worked Example: Differential Equation

Given: dy/dx = 3x^2 - 6x + 2
Analyze the differential equation and determine its critical points and behavior.
Plot the critical points on the phase plane and analyze their stability.
Draw representative solutions and analyze their stability based on the behaviors observed.

Analyzing the Differential Equation

Given: dy/dx = 3x^2 - 6x + 2
This is a first-order linear differential equation.
The highest order of the derivative is 1.
It is a polynomial equation with degree 2.
We can solve this equation using various methods.

Finding Critical Points

To find critical points, set dy/dx equal to zero:
- 3x^2 - 6x + 2 = 0
Solving this quadratic equation, we find two critical points.
Determine the values of x for which dy/dx = 0.

Plotting Critical Points

Plot the critical points on the phase plane.
Label the critical points accordingly.
Critical points are essential for understanding the behavior of the solution.

Analyzing Behavior near Critical Points

Linearize the differential equation to analyze the behavior near critical points.
Evaluate the linearized equation at each critical point.
Determine if the point is stable, unstable, or semi-stable.
This information helps understand local behavior.

Determining Asymptotic Behavior

Analyze the long-term behavior of the solution.
Determine if the solution approaches any specific values.
Check if the solution tends to infinity or zero asymptotically.
Asymptotic behavior provides insight into the overall behavior.

Drawing Representative Solutions

Choose suitable initial conditions to draw representative solutions.
Solve the differential equation using these initial conditions.
Sketch the solution curves on the phase plane.
Draw multiple curves to analyze the overall behavior.

Analyzing Stability

Analyze the stability of solutions based on their behavior and critical points.
Determine if the solution is stable, unstable, or semi-stable.
Stable solutions approach critical points.
Unstable solutions diverge from critical points.

Example: Sketching Phase Curves

Given: dy/dx = 3x^2 - 6x + 2
Analyze the equation and determine critical points and behaviors.
Plot critical points on the phase plane and analyze stability.
Draw representative solutions and analyze their stability based on behaviors.

Worked Example: Differential Equation

Given: dy/dx = 2x^2 - 4x + 1
Analyze the differential equation and determine critical points and behaviors.
Plot the critical points on the phase plane and analyze their stability.
Draw representative solutions and analyze their stability based on behaviors.

Analyzing Stability

Stable solutions approach critical points.
Unstable solutions diverge from critical points.
Semi-stable solutions exhibit a combination of stability and instability.
Analyzing stability helps understand the overall dynamics of the system.

Slide 21

The method of sketching phase curves involves analyzing the differential equation, finding critical points, plotting them on the phase plane, analyzing behavior near critical points, determining asymptotic behavior, drawing representative solutions, and analyzing stability.
Let’s apply this method to an example.

Slide 22

Example: Consider the first-order nonlinear differential equation dy/dx = x^2 - 2x.
Analyze the equation and determine its critical points and behavior.
Plot the critical points on the phase plane and analyze their stability.
Draw representative solutions and analyze their stability based on behaviors.

Slide 23

Analyzing the equation: dy/dx = x^2 - 2x
No critical points exist in this equation.
As there are no critical points, there won’t be any critical behaviors.

Slide 24

Determine the asymptotic behavior of the solution.
As x approaches infinity, x^2 - 2x approaches infinity.
As x approaches negative infinity, x^2 - 2x approaches negative infinity.
The solution approaches positive or negative infinity as x goes to infinity or negative infinity, respectively.

Slide 25

Drawing representative solutions:
- Choosing initial conditions: y(x = 0) = 1, y(x = 2) = 3, y(x = -2) = -1.
- Solving the differential equation with these initial conditions.
- Sketching the solution curves on the phase plane.

Slide 26

Analyzing stability:
- As the solution curves move away from the critical points, they diverge.
- The solutions are unstable.
- No stable or semi-stable behaviors exist in this equation.

Slide 27

Example: Consider the second-order linear differential equation d^2y/dx^2 + 4y = 0.
Analyze the equation and determine its critical points and behavior.
Plot the critical points on the phase plane and analyze their stability.
Draw representative solutions and analyze their stability based on behaviors.

Slide 28

Analyzing the equation: d^2y/dx^2 + 4y = 0
It is a second-order linear differential equation.
The highest order of the derivative is 2.
The equation represents simple harmonic motion or oscillatory behavior.

Slide 29

Finding critical points:
- Convert the second-order equation to a first-order system by introducing a new variable.
- Let z = dy/dx.
- Now, the equation becomes dz/dx + 4y = 0.

Slide 30

Solving the equation dz/dx + 4y = 0 gives the critical points:
- z = 0, y = 0
- This represents the equilibrium position.
The phase plane for this equation will be a collection of oscillatory curves around the equilibrium point. Greetings! Here are slides 21 to 30 for your lecture on “Differential Equations - Sketching Phase Curves.” I hope you find them helpful for your teaching.