10.4 - Examples of Direct Integration
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Direct Integration Example 1
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Let's solve the PDE using direct integration. Consider the equation ∂z/∂x = 2x + y. Who can tell me what our first step will be?
We need to integrate with respect to x, treating y as a constant!
Exactly! So when we integrate ∫(2x + y) dx, what do we get?
We get z = x² + xy + φ(y)!
Correct! Remember, φ(y) represents an arbitrary function of y that acts like a constant during integration with respect to x. Let's recap: integrating gives us a function and an arbitrary term.
Direct Integration Example 2
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Now, let’s explore another example: ∂z/∂y = x²y + y³. What’s our approach here?
We integrate with respect to y this time, treating x as a constant.
Correct! What do we obtain after integrating?
We have z = (1/2)x²y² + (1/4)y⁴ + ψ(x)!
Well done! Notice how we introduced ψ(x) as the arbitrary function of x. This emphasizes the generality of integration in PDE solutions. Let's summarize what’s vital in this integration process.
Solving with Two Partials Given
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For our last example, we’re given both ∂z/∂x = x + y and ∂z/∂y = x - y. What should we do first?
First, we integrate the first equation with respect to x!
Great! And what do we yield here?
We get z = (1/2)x² + xy + φ(y)!
Perfect! Now, who can tell me what to do next?
We differentiate z with respect to y, then equate it to the second equation!
Absolutely right! After differentiating, we can find φ'(y) by comparing the results. Let's summarize the importance of integrating and differentiating when working with multiple variables.
Introduction & Overview
Read summaries of the section's main ideas at different levels of detail.
Quick Overview
Standard
The section delves into specific examples demonstrating how to solve partial differential equations (PDEs) through direct integration. It showcases different scenarios where integration is applied to find solutions and highlights the significance of arbitrary functions.
Detailed
Examples of Direct Integration
Direct integration is a powerful tool for solving partial differential equations (PDEs), particularly those of first order. This section provides specific examples, demonstrating different applications of this method in various contexts. The first example illustrates the integration of a PDE with respect to one variable, where arbitrary functions arise as integration constants. The second example focuses on a PDE involving another variable and how arbitrary functions are treated similarly. The final example showcases the process of solving a PDE when both partial derivatives are provided, stressing the importance of differentiating the integrated outcome to reveal more about the general solution. Understanding these examples is crucial in mastering direct integration as a method for PDEs, laying the groundwork for more complex techniques used in mathematical modeling.
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Example 1: Integration with Respect to x
Chapter 1 of 3
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Chapter Content
Example 1:
∂𝑧
= 2𝑥 +𝑦
∂𝑥
Solution: Integrate with respect to 𝑥:
𝑧 = ∫(2𝑥+ 𝑦)𝑑𝑥 = 𝑥² + 𝑥𝑦 + 𝜙(𝑦)
𝜙(𝑦) is an arbitrary function of 𝑦.
Detailed Explanation
In this example, we're given a first-order PDE: the partial derivative of z with respect to x equals 2x + y. To solve it using direct integration, we perform the following steps:
1. We treat y as a constant and integrate the function (2x + y) with respect to x.
2. The integral of 2x is x², and the integral of y is xy.
3. After integration, we add an arbitrary function of y, denoted as 𝜙(y), since we are integrating with respect to x and y acts as a constant.
This process helps us find a general solution for z in terms of x and y.
Examples & Analogies
Imagine you're determining the height of water in a tank as it fills up over time. The rate of height increase (∂z/∂x) depends on how fast water flows in (2x + y). By integrating this rate, you're effectively figuring out the total height of water (z) after a certain period, while accounting for other variables like the constant flow from a garden hose (which is represented by y).
Example 2: Integration with Respect to y
Chapter 2 of 3
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Chapter Content
Example 2:
∂𝑧
= 𝑥²𝑦 +𝑦³
∂𝑦
Solution: Integrate with respect to 𝑦:
𝑧 = ∫(𝑥²𝑦 + 𝑦³)𝑑𝑦 = 𝑥²𝑦² + rac{1}{4}𝑦⁴ + 𝜓(𝑥)
𝜓(𝑥) is an arbitrary function of 𝑥.
Detailed Explanation
Here, we're given a different first-order PDE: the partial derivative of z with respect to y equals x²y + y³. The steps to solve it are as follows:
1. Treat x as a constant and integrate the function (x²y + y³) with respect to y.
2. The integral of x²y is (1/2)x²y², and the integral of y³ is (1/4)y⁴.
3. We combine these results to obtain the expression for z and add an arbitrary function of x, denoted as 𝜓(x). This step accounts for the fact that z could vary with x independently of y.
Examples & Analogies
Think of a farmer who is trying to determine the yield (z) of crops based on the amount of sunlight and rain. The yield depends on both sunlight (x²y) and other factors like rainfall (y³). As the farmer integrates the effects of rainfall over the growing season, they find the overall yield while considering the stable conditions (represented by the arbitrary function 𝜓(x)) that affect the crops.
Example 3: Utilizing Both Partials
Chapter 3 of 3
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Example 3 (Two partials given):
∂𝑧 ∂𝑧
= 𝑥+ 𝑦, = 𝑥− 𝑦
∂𝑥 ∂𝑦
Step 1: Integrate the first equation:
𝑧 = rac{𝑥²}{2} + 𝑥𝑦 + 𝜙(𝑦)
Step 2: Differentiate this with respect to 𝑦:
∂𝑧
= 𝑥 + 𝜙′(𝑦)
∂𝑦
Compare with the second equation:
𝑦²
𝑥 + 𝜙′(𝑦) = 𝑥 − 𝑦 ⇒ 𝜙′(𝑦) = −𝑦 ⇒ 𝜙(𝑦) = − rac{𝑦²}{2}
Final Answer:
𝑧 = rac{𝑥²}{2} + 𝑥𝑦 − rac{𝑦²}{2}
Detailed Explanation
In this example, we’re given two equations involving partial derivatives of z, indicating that z is connected to both x and y:
1. From the first equation, we integrate with respect to x while treating y as constant, resulting in an expression for z that includes an arbitrary function of y, 𝜙(y).
2. Next, we differentiate this expression with respect to y to find the corresponding equation, which gives us a new relation involving 𝜙′(y).
3. We compare this to the second partial derivative equation to find 𝜙(y). After solving for 𝜙(y), we substitute it back into our expression for z to find the complete solution, allowing us to understand how z behaves under the influence of both x and y.
Examples & Analogies
Consider a business analyzing profits based on varying production rates (x) and market demand (y). The profit equation depends on both production and demand. By integrating profit margins for different levels of production, then analyzing how changes in market demand affect overall profits lead to a complete understanding of not just how production impacts profits, but also how fluctuations in demand interconnect with production to ultimately shape the business's profitability.
Key Concepts
-
Direct Integration: A method for directly integrating PDEs to find solutions.
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Arbitrary Functions: Functions representing constants of integration that are dependent on other variables.
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Integration Process: The procedure of finding integrals with respect to one variable while treating others as constants.
Examples & Applications
Example 1: Solve ∂z/∂x = 2x + y, resulting in z = x² + xy + φ(y).
Example 2: Solve ∂z/∂y = x²y + y³, yielding z = (1/2)x²y² + (1/4)y⁴ + ψ(x).
Example 3: The system with both ∂z/∂x = x + y and ∂z/∂y = x - y requires integration of the first and differentiating to compare with the second.
Memory Aids
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Rhymes
When you integrate and find your way, don’t forget to add an arbitrary sway!
Stories
Imagine a mathematician in a forest of variables, where every tree represents a unique function. Every time they climb a mountain of integration, they write an arbitrary note to remember their journey!
Memory Tools
I - Integrate, A - Add arbitrary function, D - Differentiate, S - Solve. 'I A D S for PDEs!'
Acronyms
PDE
Please Direct Everything for Integration!
Flash Cards
Glossary
- Partial Differential Equation (PDE)
A mathematical equation that involves partial derivatives of multivariable functions used to model phenomena in various fields.
- Direct Integration
A method of solving PDEs by integrating the partial derivatives step-by-step with respect to corresponding variables.
- Arbitrary Function
A function added to the solution of a PDE representing constants of integration relative to other variables.
- Integrable Functions
Functions for which an integral can be computed within a given range, allowing for direct integration.
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