Partial Differential Equations - 6 | 6. Charpit’s Method | Mathematics - iii (Differential Calculus) - Vol 2
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6 - Partial Differential Equations

Practice

Interactive Audio Lesson

Listen to a student-teacher conversation explaining the topic in a relatable way.

Introduction to Charpit's Method

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0:00
Teacher
Teacher

Today, we’re diving into Charpit's Method, a systematic technique for solving first-order non-linear partial differential equations. Can anyone tell me what a partial differential equation is?

Student 1
Student 1

Isn't it an equation that contains partial derivatives of a function?

Teacher
Teacher

Exactly! We use Charpit's Method when our PDE looks like this: 𝐹(𝑥,𝑦,𝑧,𝑝,𝑞) = 0. In this case, 𝑝 and 𝑞 are the partial derivatives of 𝑧 with respect to 𝑥 and 𝑦. Remember this acronym, FPG: Function, Partial derivatives, Given equation. It refers to our starting point.

Student 2
Student 2

What makes Charpit's Method special compared to other methods?

Teacher
Teacher

Great question! Charpit’s Method is particularly useful when other methods fail, like the method of characteristics. It converts our PDE into a system of ordinary differential equations, simplifying our work.

Student 3
Student 3

So, we’re breaking it down into simpler parts!

Teacher
Teacher

Exactly! You've got it! To remember this method, think of writing down the Charpit’s equations as a ‘step-by-step recipe.’ It's crucial for solving our PDEs.

Student 4
Student 4

Can we see how to apply this in a real problem?

Teacher
Teacher

Absolutely, that’ll be coming up shortly! Let’s summarize: Charpit's method is used when direct methods fail and involves breaking down the PDE step-by-step using auxiliary equations.

Steps to Solve a PDE Using Charpit's Method

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0:00
Teacher
Teacher

Now that we understand the importance of Charpit's Method, let's go through the specific steps involved. Who can tell me what our first step is?

Student 1
Student 1

Start with the given PDE?

Teacher
Teacher

Exactly! Step one is to start with our PDE, 𝐹(𝑥,𝑦,𝑧,𝑝,𝑞) = 0. Next, we compute the necessary partial derivatives. Can anyone list these for me?

Student 2
Student 2

We need the partial derivatives with respect to 𝑥, 𝑦, 𝑧, 𝑝, and 𝑞.

Teacher
Teacher

Correct! Next, we use these derivatives to write down Charpit's auxiliary equations, which is basically a transformation of our PDE into a system of ODEs. Remember, we write them in the form: 𝑑𝑥 / ... = ... etc. Can anyone tell me what we gain by doing this?

Student 3
Student 3

We can solve the system more easily!

Teacher
Teacher

Right! Once we have the system, we solve it and ultimately find our functions 𝑝 and 𝑞. Finally, we integrate to find the complete integral or the general solution, 𝑧(𝑥,𝑦).

Student 4
Student 4

Can we summarize that process again?

Teacher
Teacher

Yes, let’s recap: Start with the PDE, compute the derivatives, write Charpit's auxiliary equations, solve the system, and integrate to find the solution.

Introduction & Overview

Read a summary of the section's main ideas. Choose from Basic, Medium, or Detailed.

Quick Overview

Charpit's Method is a systematic approach to solving first-order non-linear PDEs.

Standard

This section introduces Charpit's Method, a powerful technique for solving first-order non-linear partial differential equations. The method converts a PDE into a system of ordinary differential equations, simplifying the solution process.

Detailed

Detailed Summary of Charpit’s Method

Charpit’s Method is a powerful technique for solving first-order non-linear partial differential equations (PDEs) of the form 𝐹(𝑥,𝑦,𝑧,𝑝,𝑞) = 0, where 𝑧 = 𝑧(𝑥,𝑦), 𝑝 = ∂𝑧/∂𝑥, and 𝑞 = ∂𝑧/∂𝑦. The method is particularly valuable when the PDE cannot be easily approached with standard methods like the method of characteristics.

Objectives

  • To find the complete integral of a first-order non-linear PDE.
  • To convert the PDE into a system of ordinary differential equations (ODEs) using auxiliary equations for ease of solution.

Charpit’s Equations

For a given first-order PDE, the method evolves into five differential equations. Listing these equations provides a pathway to derive the general or complete integral of the PDE.

Steps to Solve

  1. Start with the PDE, determine partial derivatives of 𝐹.
  2. Use those derivatives to define Charpit's auxiliary equations.
  3. Solve this system of ODEs.
  4. Integrate to derive the complete integral (general solution) 𝑧(𝑥,𝑦).

With these steps and an example problem, students can visualize the method's application, simplifying complex PDEs into manageable systems.

Youtube Videos

But what is a partial differential equation?  | DE2
But what is a partial differential equation? | DE2

Audio Book

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Introduction to Charpit's Method

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Charpit’s Method is a powerful and systematic technique used to solve first-order non-linear partial differential equations (PDEs) of the form:

𝐹(𝑥,𝑦,𝑧,𝑝,𝑞) = 0

where
• 𝑧 = 𝑧(𝑥,𝑦) is the unknown function,
• 𝑝 = ∂𝑧/∂𝑥
• 𝑞 = ∂𝑧/∂𝑦

This method was introduced by French mathematician Jean Charpit and is particularly useful when a PDE is not easily solvable through standard methods like method of characteristics or Lagrange’s method.

Detailed Explanation

Charpit’s Method is essentially a strategy for addressing complex problems in mathematics, specifically those involving partial differential equations (PDEs). These equations are 'partial' because they involve functions of multiple variables and their partial derivatives. The method provides a structured approach to transforming a PDE into a format that can be easier to work with, particularly when traditional methods do not yield results.

Examples & Analogies

Think of Charpit’s Method like having a complicated recipe with multiple steps and ingredients. Traditional methods might not give you the right mixture if steps are skipped or if there are too many components to handle at once. Charpit’s Method helps break down and organize these steps, similar to how a good chef would outline a process in order to make a complicated dish easier to follow.

Objectives of Charpit’s Method

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• To find complete integral of a first-order non-linear PDE.
• To convert the PDE into a system of ordinary differential equations (ODEs) using auxiliary equations, which can be solved to obtain the solution.

Detailed Explanation

The main goals of Charpit’s Method are twofold. First, it aims to find the 'complete integral' of a given PDE, which is essentially the general solution that fits the equation. Second, it seeks to reframe the PDE into a simpler form: a system of ordinary differential equations. These ODEs are one-dimensional and generally easier to solve, allowing us to ultimately derive the solution to the original PDE.

Examples & Analogies

Consider trying to solve a complex puzzle. Your objective is to see the entire picture (the complete integral) but the pieces you have are scattered. Charpit's Method helps you to sort these pieces into smaller, simpler groups (ODEs), making it easier to see how they fit together to form the complete image.

Understanding Charpit’s Equations

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Given a first-order PDE:

𝐹(𝑥,𝑦,𝑧,𝑝,𝑞) = 0

Define:
• 𝑝 = ∂𝑧/∂𝑥
• 𝑞 = ∂𝑧/∂𝑦

Then, Charpit's equations are:

𝑑𝑥/∂𝐹/∂𝑝 = 𝑑𝑦/∂𝐹/∂𝑞 = 𝑑𝑧/(𝑝⋅∂𝐹/∂𝑝 + 𝑞⋅∂𝐹/∂𝑞 − ∂𝐹/∂𝑥 − 𝑝⋅∂𝐹/∂𝑧)

𝑑𝑞/−∂𝐹/∂𝑦 − 𝑞⋅∂𝐹/∂𝑧

This gives us a system of five differential equations, and the solution of this system provides us with the general or complete integral of the PDE.

Detailed Explanation

Charpit's equations create a link between the variables of the PDE and their relationships. By taking derivatives of the function F with respect to its variables, we craft a series of equations that can be handled more straightforwardly than the original PDE. Solving these equations gives insights into the behavior of the unknown function z regarding the dependent variables x and y, ultimately leading to the solution.

Examples & Analogies

Imagine trying to map out how a river flows through a landscape. The Charpit equations represent the flow and direction of the water at various points. By solving these equations, similar to how hydrologists do, we can predict the water's path and behavior, leading us to an understanding of the overall river system (the solution to the PDE).

Steps to Solve a PDE Using Charpit’s Method

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  1. Start with the given PDE: 𝐹(𝑥,𝑦,𝑧,𝑝,𝑞) = 0
  2. Calculate the partial derivatives:
    • 𝐹𝑥 = ∂𝐹/∂𝑥
    • 𝐹𝑦 = ∂𝐹/∂𝑦
    • 𝐹𝑧 = ∂𝐹/∂𝑧
    • 𝐹𝑝 = ∂𝐹/∂𝑝
    • 𝐹𝑞 = ∂𝐹/∂𝑞
  3. Write the Charpit's auxiliary equations using:
    𝑑𝑥/𝑓 𝑑𝑦/𝑓 𝑑𝑧/𝑓 = ... (as derived)
  4. Solve the system of ODEs using any possible combination of equations.
  5. Find the expressions for 𝑝 and 𝑞 in terms of 𝑥,𝑦,𝑧.
  6. Integrate to obtain the complete integral (general solution) 𝑧(𝑥,𝑦).

Detailed Explanation

To solve a PDE using Charpit's Method, one follows a systematic set of steps. The process starts with the existing PDE and its derivatives are computed. This leads to the formulation of Charpit's auxiliary equations, which consist of multiple ordinary differential equations. These are then solved to find values for p and q, which are eventually used to get a complete integral for z, the solution we initially sought.

Examples & Analogies

Think of the steps like following a guide to assemble furniture. You start with the box (the PDE), determine which parts you have (the derivatives), follow instructions (auxiliary equations), piece things together (solve ODEs), and finally, you get a chair or table (the complete integral). Each step is crucial to make sure you end up with the right product at the end.

Example Problem Using Charpit’s Method

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Solve the PDE using Charpit’s Method:
𝑧 = 𝑝𝑥 +𝑞𝑦 +𝑝𝑞
Step 1: Convert to standard form
Bring all terms to one side:
𝐹(𝑥,𝑦,𝑧,𝑝,𝑞) = 𝑝𝑥 +𝑞𝑦+ 𝑝𝑞 −𝑧 = 0
Step 2: Compute partial derivatives
• 𝐹𝑥 = 𝑥 + 𝑞/𝑝
• 𝐹𝑦 = 𝑦 + 𝑝/𝑞
• 𝐹𝑧 = 𝑝
• 𝐹𝑝 = 𝑞
• 𝐹𝑞 = −1
Step 3: Write Charpit’s equations
Substitute: 𝑑𝑥/𝑓 = ...

So,
• ⇒ 𝑝 = 𝑎 (a constant)
• ⇒ 𝑞 = 𝑏 (a constant)
Step 4: Use p and q in original equation
Substitute 𝑝 = 𝑎, 𝑞 = 𝑏 in the original equation:
𝑧 = 𝑎𝑥 + 𝑏𝑦 + 𝑎𝑏
This is the complete integral (general solution).

Detailed Explanation

In this example, the PDE given is manipulated step-by-step. It begins by converting it into a standard form. Partial derivatives are then calculated one by one to feed into Charpit’s method. The connections formed in the auxiliary equations lead to constants p and q, which ultimately allow us to find z in relation to x and y in a complete integral form.

Examples & Analogies

Consider a detective piecing together a mystery. Each clue (the steps and derivatives) leads them closer to the truth (the complete integral). By following these leads in a logical sequence, they can finally reveal the mystery of what happened (the solution to the PDE).

Graphical Interpretation of Charpit’s Method

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Charpit’s method geometrically corresponds to finding characteristic curves in the 5D space of variables (𝑥,𝑦,𝑧,𝑝,𝑞) along which the PDE reduces to an ODE.

Detailed Explanation

In the context of geometry, Charpit’s Method can be visualized as identifying specific curves in a five-dimensional space defined by the variables of the problem. These curves represent paths along which the relationships described by the PDE simplify; transforming them into ordinary differential equations makes them easier to analyze and understand.

Examples & Analogies

Imagine navigating through a maze (the five-dimensional space). Finding the right pathways that lead to straight shots (the characteristic curves) simplifies your journey, allowing you to navigate the maze faster by avoiding complicated turns (the complex nature of the PDE) and instead simplifying your route into simpler sections (the ODEs).

Summary of Charpit’s Method

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• Charpit’s method is a systematic way to solve first-order non-linear PDEs of the form 𝐹(𝑥,𝑦,𝑧,𝑝,𝑞) = 0.
• It converts the PDE into five auxiliary ODEs using partial derivatives of 𝐹.
• By solving this system, we obtain values of 𝑝, 𝑞, and ultimately the solution 𝑧(𝑥,𝑦).
• The method works well for PDEs that are non-linear in 𝑝 and 𝑞, where other techniques may fail.

Detailed Explanation

The summary encapsulates the core principles of Charpit’s Method, emphasizing its systematic nature when addressing first-order non-linear PDEs. It highlights how the method restructures the problem into a more manageable form (the five auxiliary ODEs) to derive the solution effectively.

Examples & Analogies

Think of a toolkit designed for solving specific problems. Charpit's Method acts as that toolkit for tackling certain types of complex equations. Just as some tools are specialized for different tasks (like carpentry or plumbing), Charpit's Method is designed for those tricky scenarios where other methods might not suffice.

Definitions & Key Concepts

Learn essential terms and foundational ideas that form the basis of the topic.

Key Concepts

  • Charpit's Method: A technique for simplifying and solving PDEs.

  • Auxiliary Equations: Used to convert PDEs into ODE systems.

  • Complete Integral: The ultimate goal of applying Charpit's Method.

Examples & Real-Life Applications

See how the concepts apply in real-world scenarios to understand their practical implications.

Examples

  • Using Charpit’s Method on a non-linear PDE, one can derive the general solution, which includes constants of integration from ODEs.

  • Transforming the equation 𝐹(𝑥,𝑦,𝑧,𝑝,𝑞) = 0 offers a clear structure for solution and allows for solving via well-known techniques.

Memory Aids

Use mnemonics, acronyms, or visual cues to help remember key information more easily.

🎵 Rhymes Time

  • When the PDE’s at play, Charpit's Method saves the day!

📖 Fascinating Stories

  • Imagine a detective named Charpit who uses a special code to decipher clues in a complex puzzle, helping him solve the mystery step-by-step, leading to the truth.

🧠 Other Memory Gems

  • Remember 'SDEI' for steps: Start with PDE, Derivatives, Equations, Integrate.

🎯 Super Acronyms

PGS - Partial, Given, Solve! Remember to identify the PDE, what is given, and how you solve it.

Flash Cards

Review key concepts with flashcards.

Glossary of Terms

Review the Definitions for terms.

  • Term: Partial Differential Equation (PDE)

    Definition:

    An equation containing unknown multivariable functions and their partial derivatives.

  • Term: Charpit’s Method

    Definition:

    A systematic technique for solving first-order non-linear PDEs by turning them into a system of ordinary differential equations.

  • Term: Ordinary Differential Equation (ODE)

    Definition:

    A differential equation containing one or more functions of one independent variable and its derivatives.

  • Term: Auxiliary Equations

    Definition:

    Equations derived from a PDE to facilitate finding solutions.

  • Term: Complete Integral

    Definition:

    The general solution of a PDE that encompasses all possible solutions.