First Order PDE
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Table of Contents
Finding the Complete Integral
Problem: 3
Find the complete integral of the partial differential equation \[ \Big(\frac{\partial z}{\partial x}+\frac{\partial z}{\partial y}\Big)\Big(z-x\frac{\partial z}{\partial x}-y\frac{\partial z}{\partial y}\Big)=1\]
Step 1: Restate the Problem
We wish to find the complete integral (a two-parameter family of solutions) of the partial differential equation \[ \Big(z_x + z_y\Big)\Big(z – x\,z_x – y\,z_y\Big) = 1, \] where \( z_x = \frac{\partial z}{\partial x} \) and \( z_y = \frac{\partial z}{\partial y} \).
Step 2: Key Observation
Notice that the expression \[ z – x\,z_x – y\,z_y \] suggests that if we try a solution of the form of a plane, \[ z(x,y) = ax + by + c, \] where \(a\), \(b\), and \(c\) are constants, then the derivatives are \begin{align} z_x &= a, \quad \text{and} \quad z_y = b. \tag{1} \end{align} Thus, \begin{align} z_x + z_y &= a + b, \tag{2} \end{align} and \begin{align} z – x\,z_x – y\,z_y &= (ax+by+c) – x\,a – y\,b = c. \tag{3} \end{align}
Step 3: Substitution into the PDE
Substituting the expressions from (2) and (3) into the PDE, we obtain: \begin{align} (a+b)\,c &= 1. \tag{4} \end{align} Equation (4) tells us that to satisfy the PDE, the constant \(c\) must be chosen as \begin{align} c &= \frac{1}{a+b}, \quad \text{with} \quad a+b \neq 0. \tag{5} \end{align}
Step 4: The Complete Integral
Hence, the complete integral of the PDE is given by: \begin{align} z(x,y) = ax + by + \frac{1}{a+b}, \quad \text{with} \quad a+b \neq 0. \tag{6} \end{align}
Summary
We assumed a plane solution of the form \(z(x,y) = ax + by + c\). Calculating the derivatives led to the relation \((a+b)c = 1\). By choosing \[ c = \frac{1}{a+b}, \] we obtained the complete integral \[ z(x,y) = ax + by + \frac{1}{a+b}, \quad a+b\neq0, \] which indeed satisfies the given PDE.
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FAQs
Partial Differential Equations
- What is a partial differential equation (PDE)?
A PDE is an equation that involves unknown multivariable functions and their partial derivatives. It describes how the function changes with respect to multiple independent variables.
- How do PDEs differ from ordinary differential equations (ODEs)?
Unlike ODEs, which involve derivatives with respect to a single variable, PDEs involve partial derivatives with respect to two or more independent variables.
- What are the common types of PDEs?
PDEs are generally classified into three types based on their characteristics:
- Elliptic: e.g., Laplace’s equation
- Parabolic: e.g., the heat equation
- Hyperbolic: e.g., the wave equation
- What role do boundary and initial conditions play?
- Boundary conditions specify the behavior of the solution along the edges of the domain.
- Initial conditions are used in time-dependent problems to define the state of the system at the start.
- What methods are commonly used to solve PDEs?
There are several techniques, including:
- Analytical methods like separation of variables, Fourier and Laplace transforms, and the method of characteristics
- Numerical methods such as finite difference, finite element, and spectral methods
- What is the method of separation of variables?
This method assumes that the solution can be written as a product of functions, each depending on only one of the independent variables. This assumption reduces the PDE to a set of simpler ODEs that can be solved individually.
- In which fields are PDEs applied?
PDEs model a wide range of phenomena across various fields including physics (heat transfer, fluid dynamics), engineering (stress analysis, electromagnetics), finance (option pricing models), and more.
- What distinguishes linear from nonlinear PDEs?
- Linear PDEs have terms that are linear with respect to the unknown function and its derivatives, making them more tractable analytically.
- Nonlinear PDEs include terms that are nonlinear, often leading to complex behaviors and requiring specialized methods for solution.
- How do you determine the order of a PDE?
The order of a PDE is defined by the highest derivative (partial derivative) present in the equation. For example, if the highest derivative is a second derivative, the PDE is second order.
- What are some common challenges in solving PDEs?
Challenges include finding closed-form analytical solutions, handling complex geometries and boundary conditions, and the significant computational effort required for accurate numerical solutions.
