Given that x^2+y^2 solves the Poisson equation (∂^2 u)/(∂x^2)+(∂^2 u)/(∂y^2)=4, while x^4+y^4 so

Given that x^2+y^2 solves the Poisson equation (∂^2...

Key Concepts

Particular Solutions

A particular solution of a differential equation is a specific solution that satisfies the given non-homogeneous equation, without necessarily satisfying the associated homogeneous boundary conditions. In the context of the Poisson equation, finding a particular solution often involves targeting the structure of the source term f and, as in this example, using tailored functions whose Laplacians yield desired forms such as constants or quadratic expressions.

Linearity and Superposition Principle

Linearity in the context of differential equations means that the equation obeys the rules of additivity and scalar multiplication. For the Poisson equation, this allows one to form a particular solution for a composite source by adding particular solutions corresponding to the individual components of the source. This principle of superposition is used to combine known solutions for parts of the forcing function into a single solution that addresses the whole problem.

Poisson Equation

The Poisson equation is a partial differential equation of the form ?²u = f, where ?² is the Laplacian operator and f is a given function. It arises in many fields such as physics and engineering, particularly in contexts like electrostatics and heat flow. In solving a Poisson equation, one typically seeks a function u whose second derivatives sum to match the source function f.

Laplacian Operator

The Laplacian operator, denoted by ?², is a second-order differential operator defined as the sum of the second partial derivatives with respect to each spatial variable. In two dimensions, it is given by ?²u = uxx + uyy. This operator is central to equations describing diffusion, wave propagation, and potential fields, and its properties allow for techniques like superposition in linear PDEs.

Transcript

00:01 Good day.

00:01 The problem we're working with today is a simple one of finding second derivatives.

00:08 Now, they're asking us to find f derived twice to x, f derived twice to y, f derived first to x and then to y, then f derived to y, then f derived to x.

00:20 Now, at the top here, you'll see i have the textbook's notation.

00:28 At the bottom, i have another notation that i roll down.

00:32 It's just a bit shorter, a bit easier to write down.

00:35 I'm just going to be using this notation for this exercise to save us a little bit of time.

00:41 I also feel that it's a bit clearer, but just keep in mind that these correspond to the ones directly above them.

00:49 Now, first we're going to start with the finding the first derivatives of x.

00:54 Now, so that's by doing, we find f.

01:00 Derived to x, which in the notation i'm using here is written like this.

01:09 And if we derive each of these to x, we get x times e to the y derived to x gives us each of the y, because x becomes one.

01:22 And then we have x to the power four times y, derived to x gives us four x cubed times y, derived and then y cubed derived to x gives us zero because the x is assumed we assume we have an x to the power zero here because it's one and then a constant derived uh yeah a constant derived gives us a zero and y cubed is a constant is a constant is a constant in this case since we're deriving to x.

02:09 Then when we get f derived to y, this gives us, in this case, we have e to the power y derived to x just gives us the exact same thing because we get the derivative of y, which is 1 times the same thing.

02:33 That's the rule when we come to e to the power, to the power of a variable.

02:40 So it's just x times e to the power y.

02:45 And then x to the power four times y derived to y just gives us x to the power four.

02:54 And then y to about three derived to y gives us three y squared.

03:01 Okay.

03:02 So then once we have these two down, it's pretty simple getting the rest.

03:08 So f derived twice to x is this derivative up here.

03:19 Just derived to x once again...