Slide 1: Differential Equations - Continuation of Comparison of Infinities

  • Recap: In the previous lecture, we discussed the concept of “Comparison of Infinities” in differential equations.
  • Today, we will continue our exploration of this topic and go deeper into the mathematical implications.

Slide 2: Comparison of Infinities - Quick Review

  • In differential equations, we often encounter situations where the behavior of a function can be determined by comparing it with another function.
  • This technique is known as the “Comparison of Infinities” and helps us analyze the solution behavior without explicitly solving the equation.
  • We use this method when dealing with higher order linear differential equations.

Slide 3: Types of Behavior

  • When comparing infinities, there are three possible outcomes:
    • Case 1: If the two functions have the same behavior, we say they are “comparable.”
    • Case 2: If one function grows at a faster rate than the other, we say they are “incomparable.”
    • Case 3: If one function dominates the other by growing much faster, we say they are “comparable with domination.”

Slide 4: Comparable Functions

  • When two functions are comparable, we can draw useful conclusions about their solutions.
  • For example, if we have a linear differential equation with a known solution, we can determine the behavior of an unknown solution by comparing the two functions. Example: Let’s consider the equation $y’’ + y = e^x$. We know that $y = e^x$ is a solution. By comparing the exponential growth of $e^x$ with the unknown solution, we can conclude that the unknown solution also grows exponentially.

Slide 5: Incomparable Functions

  • When two functions are incomparable, their behaviors do not provide direct insights into each other.
  • In such cases, solving the differential equation explicitly becomes necessary to determine the behavior of the unknown solution. Example: Consider the equation $y’’ + y = \sin(x)$. Here, the behavior of $\sin(x)$ is different from any known solution. Solving the equation using methods like variations of parameters or undetermined coefficients becomes necessary.

Slide 6: Comparable Functions with Domination

  • When one function dominates another, it implies that the dominant function determines the behavior of the other function.
  • This situation simplifies the analysis of the unknown solution since we can focus on the dominant function. Example: Let’s consider the equation $y’’ - y = e^x$. Here, $e^x$ dominates any solution that grows exponentially with a different base.

Slide 7: Comparison of Infinities - Steps

  • To effectively compare infinities, we follow these steps:
    1. Start with a given differential equation.
    2. Identify a known solution or a function that behaves similarly to the unknown solution.
    3. Compare the growth or decay rate of both functions.
    4. Determine if the functions are comparable, incomparable, or comparable with domination.
    5. Draw conclusions about the behavior of the unknown solution based on the comparison.

Slide 8: Example 1 - Comparable Functions

  • Let’s work through an example to illustrate the concept of comparable functions. Example: Consider the equation $y’’ - 4y’ + 3y = e^{2x}$. We know that $y = e^{2x}$ is a solution. By comparing the growth rate of $e^{2x}$ with the unknown solution, we can conclude that the unknown solution also grows exponentially with a base of 2.

Slide 9: Example 2 - Incomparable Functions

  • Let’s now explore an example of incomparable functions. Example: Consider the equation $y’’ + 2y’ + y = \cos(x)$. Here, the behavior of $\cos(x)$ is different from any known solution. Solving the equation explicitly becomes necessary to determine the behavior of the unknown solution.

Slide 10: Example 3 - Comparable Functions with Domination

  • Lastly, let’s examine an example of comparable functions with domination. Example: Consider the equation $y’’ + 2y’ + y = e^x$. Here, $e^x$ dominates any solution that grows exponentially with a different base. Therefore, we can conclude that the unknown solution also grows exponentially with a base of $e$.
  1. Recap: Differential Equations - Continuation of Comparison of Infinities
  • In the previous lecture, we discussed the concept of “Comparison of Infinities” in differential equations.
  • Today, we will continue our exploration of this topic and go deeper into the mathematical implications.
  1. Types of Behavior
  • When comparing infinities, there are three possible outcomes:
    • Case 1: If the two functions have the same behavior, we say they are “comparable.”
    • Case 2: If one function grows at a faster rate than the other, we say they are “incomparable.”
    • Case 3: If one function dominates the other by growing much faster, we say they are “comparable with domination.”
  1. Comparable Functions
  • When two functions are comparable, we can draw useful conclusions about their solutions.
  • For example, if we have a linear differential equation with a known solution, we can determine the behavior of an unknown solution by comparing the two functions. Example: Let’s consider the equation $y’’ + y = e^x$. We know that $y = e^x$ is a solution. By comparing the exponential growth of $e^x$ with the unknown solution, we can conclude that the unknown solution also grows exponentially.
  1. Incomparable Functions
  • When two functions are incomparable, their behaviors do not provide direct insights into each other.
  • In such cases, solving the differential equation explicitly becomes necessary to determine the behavior of the unknown solution. Example: Consider the equation $y’’ + y = \sin(x)$. Here, the behavior of $\sin(x)$ is different from any known solution. Solving the equation using methods like variations of parameters or undetermined coefficients becomes necessary.
  1. Comparable Functions with Domination
  • When one function dominates another, it implies that the dominant function determines the behavior of the other function.
  • This situation simplifies the analysis of the unknown solution since we can focus on the dominant function. Example: Let’s consider the equation $y’’ - y = e^x$. Here, $e^x$ dominates any solution that grows exponentially with a different base.
  1. Comparison of Infinities - Steps
  • To effectively compare infinities, we follow these steps:
    1. Start with a given differential equation.
    2. Identify a known solution or a function that behaves similarly to the unknown solution.
    3. Compare the growth or decay rate of both functions.
    4. Determine if the functions are comparable, incomparable, or comparable with domination.
    5. Draw conclusions about the behavior of the unknown solution based on the comparison.
  1. Example 1 - Comparable Functions
  • Let’s work through an example to illustrate the concept of comparable functions. Example: Consider the equation $y’’ - 4y’ + 3y = e^{2x}$. We know that $y = e^{2x}$ is a solution. By comparing the growth rate of $e^{2x}$ with the unknown solution, we can conclude that the unknown solution also grows exponentially with a base of 2.
  1. Example 2 - Incomparable Functions
  • Let’s now explore an example of incomparable functions. Example: Consider the equation $y’’ + 2y’ + y = \cos(x)$. Here, the behavior of $\cos(x)$ is different from any known solution. Solving the equation explicitly becomes necessary to determine the behavior of the unknown solution.
  1. Example 3 - Comparable Functions with Domination
  • Lastly, let’s examine an example of comparable functions with domination. Example: Consider the equation $y’’ + 2y’ + y = e^x$. Here, $e^x$ dominates any solution that grows exponentially with a different base. Therefore, we can conclude that the unknown solution also grows exponentially with a base of $e$.
  1. Summary
  • In this lecture, we continued our discussion on the “Comparison of Infinities” in differential equations.
  • We explored different types of behaviors that can arise when comparing infinities.
  • We learned how to analyze comparable, incomparable, and comparable-with-domination functions.
  • Through examples, we applied these concepts to determine the behavior of unknown solutions.
  • Understanding the comparison of infinities helps us gain insights into differential equations without explicitly solving them.
  1. Recap: Differential Equations - Continuation of Comparison of Infinities
  • In the previous lecture, we discussed the concept of “Comparison of Infinities” in differential equations.
  • Today, we will continue our exploration of this topic and go deeper into the mathematical implications.
  1. Types of Behavior
  • When comparing infinities, there are three possible outcomes:
    • Case 1: If the two functions have the same behavior, we say they are “comparable.”
    • Case 2: If one function grows at a faster rate than the other, we say they are “incomparable.”
    • Case 3: If one function dominates the other by growing much faster, we say they are “comparable with domination.”
  1. Comparable Functions
  • When two functions are comparable, we can draw useful conclusions about their solutions.
  • For example, if we have a linear differential equation with a known solution, we can determine the behavior of an unknown solution by comparing the two functions. Example: Let’s consider the equation $y’’ + y = e^x$. We know that $y = e^x$ is a solution. By comparing the exponential growth of $e^x$ with the unknown solution, we can conclude that the unknown solution also grows exponentially.
  1. Incomparable Functions
  • When two functions are incomparable, their behaviors do not provide direct insights into each other.
  • In such cases, solving the differential equation explicitly becomes necessary to determine the behavior of the unknown solution. Example: Consider the equation $y’’ + y = \sin(x)$. Here, the behavior of $\sin(x)$ is different from any known solution. Solving the equation using methods like variations of parameters or undetermined coefficients becomes necessary.
  1. Comparable Functions with Domination
  • When one function dominates another, it implies that the dominant function determines the behavior of the other function.
  • This situation simplifies the analysis of the unknown solution since we can focus on the dominant function. Example: Let’s consider the equation $y’’ - y = e^x$. Here, $e^x$ dominates any solution that grows exponentially with a different base.
  1. Comparison of Infinities - Steps
  • To effectively compare infinities, we follow these steps:
    1. Start with a given differential equation.
    2. Identify a known solution or a function that behaves similarly to the unknown solution.
    3. Compare the growth or decay rate of both functions.
    4. Determine if the functions are comparable, incomparable, or comparable with domination.
    5. Draw conclusions about the behavior of the unknown solution based on the comparison.
  1. Example 1 - Comparable Functions
  • Let’s work through an example to illustrate the concept of comparable functions. Example: Consider the equation $y’’ - 4y’ + 3y = e^{2x}$. We know that $y = e^{2x}$ is a solution. By comparing the growth rate of $e^{2x}$ with the unknown solution, we can conclude that the unknown solution also grows exponentially with a base of 2.
  1. Example 2 - Incomparable Functions
  • Let’s now explore an example of incomparable functions. Example: Consider the equation $y’’ + 2y’ + y = \cos(x)$. Here, the behavior of $\cos(x)$ is different from any known solution. Solving the equation explicitly becomes necessary to determine the behavior of the unknown solution.
  1. Example 3 - Comparable Functions with Domination
  • Lastly, let’s examine an example of comparable functions with domination. Example: Consider the equation $y’’ + 2y’ + y = e^x$. Here, $e^x$ dominates any solution that grows exponentially with a different base. Therefore, we can conclude that the unknown solution also grows exponentially with a base of $e$.
  1. Summary
  • In this lecture, we continued our discussion on the “Comparison of Infinities” in differential equations.
  • We explored different types of behaviors that can arise when comparing infinities.
  • We learned how to analyze comparable, incomparable, and comparable-with-domination functions.
  • Through examples, we applied these concepts to determine the behavior of unknown solutions.
  • Understanding the comparison of infinities helps us gain insights into differential equations without explicitly solving them.