1.2.2 - Example 4
Enroll to start learning
You’ve not yet enrolled in this course. Please enroll for free to listen to audio lessons, classroom podcasts and take practice test.
Interactive Audio Lesson
Listen to a student-teacher conversation explaining the topic in a relatable way.
Understanding Partial Differential Equations
🔒 Unlock Audio Lesson
Sign up and enroll to listen to this audio lesson
Today, we're diving into Partial Differential Equations, or PDEs. Can anyone tell me what a PDE is?
Isn't it an equation that involves partial derivatives?
Yeah, like the Laplace equation?
Exactly! PDEs are used in fields such as fluid dynamics and heat transfer. They describe how a quantity changes in relation to multiple independent variables. The Laplace equation you mentioned serves as a great example of a second-order PDE.
What does it mean by 'forming a PDE'?
Great question! Forming a PDE means deriving it by eliminating arbitrary constants or functions from a given equation.
Let’s remember that by thinking of PDE as 'Partial Derivative Equation'.
Eliminating Arbitrary Constants
🔒 Unlock Audio Lesson
Sign up and enroll to listen to this audio lesson
Let's start with eliminating arbitrary constants. Can someone explain how we can derive a PDE from a function that has constants?
Do we differentiate the function?
Exactly! We differentiate with respect to the independent variables. For example, if we have z = ax + by + c, we differentiate to get the values of 'a' and 'b'.
And then substitute those into the original equation, right?
Correct! And that leads to eliminating 'c' as well. The result still needs to be expressed properly, like turning it into a PDE form.
Can you summarize that process?
Certainly! Differentiate, substitute, and eliminate to form the PDE—remember: 'DSE' for differentiate, substitute, eliminate.
Eliminating Arbitrary Functions
🔒 Unlock Audio Lesson
Sign up and enroll to listen to this audio lesson
Now, let’s shift gears to eliminating arbitrary functions. How would you approach that?
We would use differentiation but on the arbitrary function?
Correct! When you have a function like z = f(x² + y²), you express it in terms of a new variable and then differentiate it.
And then we eliminate the function to get the PDE?
That’s right! Remember, when you have functions involved, carefully use the chain rule for differentiation.
What’s a key takeaway for that method?
Just remember that for arbitrary functions, we Focus on differentiation and elimination—'FDE'.
Introduction & Overview
Read summaries of the section's main ideas at different levels of detail.
Quick Overview
Standard
In this section, students learn how to form PDEs through elimination of arbitrary constants and functions from original equations. It provides a structured approach to derive PDEs using differentiation, supplemented by examples to illustrate the methods.
Detailed
Formation of Partial Differential Equations
Partial Differential Equations (PDEs) involve partial derivatives of multivariable functions and are vital in various fields including physics and engineering. The formation of a PDE is achieved by eliminating arbitrary constants or functions from a given equation, which results in an expression that is representative of an entire class of functions satisfying physical or geometric conditions. This section dives into the methods of obtaining PDEs, touching on how to eliminate constants and functions through partial differentiation, with practical examples to solidify understanding.
Key Concepts:
- What is a PDE?
- Definition and examples, including common forms such as the Laplace Equation.
- Eliminating Arbitrary Constants:
- Describes the method to differentiate and eliminate constants to arrive at the PDE.
- Eliminating Arbitrary Functions:
- Similar to constants, but using the properties of functions through the differentiation processes to derive the PDE.
By mastering these processes, students will be prepared to tackle various problems involving PDEs.
Youtube Videos
Audio Book
Dive deep into the subject with an immersive audiobook experience.
Example Overview
Chapter 1 of 4
🔒 Unlock Audio Chapter
Sign up and enroll to access the full audio experience
Chapter Content
Given:
z=f(x+y)+g(x−y)
Let:
u=x+y, v=x−y⇒z=f(u)+g(v)
Detailed Explanation
This chunk introduces a mathematical function z that depends on two new variables, u and v. Here, u is defined as (x+y) and v as (x−y). The function z is represented as a sum of two other functions, f and g, which depend on these variables u and v respectively.
Examples & Analogies
Think of z as a recipe that combines two ingredients (u and v). Each ingredient (the functions f and g) contributes uniquely to the overall flavor (the function z). Just like different combinations of ingredients can create new dishes, different combinations of u and v will change the outcome of z.
Differentiation Steps
Chapter 2 of 4
🔒 Unlock Audio Chapter
Sign up and enroll to access the full audio experience
Chapter Content
Differentiate:
∂z ∂z
=f′(u)+g′(v), =f′(u)−g′(v)
∂x ∂y
Detailed Explanation
In this step, we take the partial derivatives of z with respect to x and y. The partial derivative with respect to x leads to f'(u) + g'(v), while the derivative with respect to y gives us f'(u) - g'(v). This mathematical manipulation helps to find how z changes as we slightly change x or y, which is crucial for forming a PDE.
Examples & Analogies
Imagine monitoring the temperature of a mixture as you add different ingredients. The way the temperature changes when you add each ingredient is like taking a derivative. It tells you how sensitive the overall result (temperature) is to changes in each of the ingredients (x and y).
Adding and Subtracting Components
Chapter 3 of 4
🔒 Unlock Audio Chapter
Sign up and enroll to access the full audio experience
Chapter Content
Add and subtract:
p=f′(u)+g′(v), q=f′(u)−g′(v)
Add:
p+q
p+q=2f′(u)⇒f′(u)=
2
Subtract:
p−q
p−q=2g′(v)⇒g′(v)=
2
Detailed Explanation
This chunk elaborates on the manipulation of the derivatives we calculated earlier. By adding and subtracting these equations, we can isolate the derivatives of f and g. The results show that f' and g' can be defined in simpler forms, expressed in terms of p and q.
Examples & Analogies
Consider a situation where you’re measuring the height of a plant from two different perspectives. If you add the heights from both perspectives, you get a combined view. By doing so, you can also find out the individual heights (f’ and g’) of the different parts of the plant just like we’re finding those derivatives here.
Eliminating Functions to Form PDE
Chapter 4 of 4
🔒 Unlock Audio Chapter
Sign up and enroll to access the full audio experience
Chapter Content
Now differentiate again and eliminate f and g to form a second-order PDE.
Detailed Explanation
Finally, we take one more derivative step, which allows us to eliminate the arbitrary functions f and g using the relationships we've established. This leads us toward a second-order Partial Differential Equation (PDE) that describes the changes in z based on the variables x and y without dependence on arbitrary functions.
Examples & Analogies
Imagine you’ve gathered enough data on a plant’s growth (the function z) that you no longer need to refer to detailed observations (the functions f and g). You’ve formed a general rule (the PDE) that predicts future growth based on the current conditions, simplifying your analysis.
Key Concepts
-
What is a PDE?
-
Definition and examples, including common forms such as the Laplace Equation.
-
Eliminating Arbitrary Constants:
-
Describes the method to differentiate and eliminate constants to arrive at the PDE.
-
Eliminating Arbitrary Functions:
-
Similar to constants, but using the properties of functions through the differentiation processes to derive the PDE.
-
By mastering these processes, students will be prepared to tackle various problems involving PDEs.
Examples & Applications
z = ax + by + c leads to a PDE by differentiating with respect to x and y.
z = f(x^2 + y^2) leads to a PDE after substituting and eliminating f.
Memory Aids
Interactive tools to help you remember key concepts
Rhymes
To form a PDE, differentiate away, substitute, then remove in a smart way.
Stories
Imagine a mathematician on a mountain, trying to find patterns in nature; by removing distractions, the beauty of PDEs revealed itself.
Memory Tools
Remember 'FDE' to focus on Functions, Differentiation, and Elimination when dealing with arbitrary functions.
Acronyms
DSE = Differentiate, Substitute, Eliminate for dealing with constants.
Flash Cards
Glossary
- Partial Differential Equation (PDE)
An equation involving partial derivatives of a multivariable function.
- Arbitrary Constants
Values in an equation that can be altered without changing the fundamental nature of the equation.
- Arbitrary Functions
Functions that contain parameters which can vary without losing their validity.
- Differentiation
The process of obtaining the derivative of a function.
Reference links
Supplementary resources to enhance your learning experience.