X w^''-w^'-x w=0 | Numerade

Name: Problem 27, Chapter 8: Fundamentals of Differential Equations
Uploaded: 2020-07-10T13:36:39-08:00
Duration: 7 min 25 s
Channel: Jack Chen
Description: x w^''-w^'-x w=0

Key Concepts

Ordinary Differential Equations (ODEs)

An ordinary differential equation involves functions of one independent variable and their derivatives. Understanding ODEs is fundamental in mathematics, physics, and engineering because they describe a wide range of dynamic phenomena. They serve as a basic model for systems that change continuously over time or space.

Linear Homogeneous Differential Equations

A linear homogeneous differential equation is one in which the dependent variable and all its derivatives appear linearly and the equation is set equal to zero. These equations have the important property that any linear combination of solutions is also a solution, making them amenable to powerful superposition techniques and methods for finding the general solution.

Variable Coefficient Differential Equations

Variable coefficient differential equations have coefficients that are functions of the independent variable, rather than constants. This variation often makes them more challenging to solve analytically and may require special techniques such as transformation methods or series solutions. Their study is crucial since many real-world systems involve non-constant parameters.

Singular Points

A singular point of a differential equation is a value of the independent variable where the behavior of the solution may change dramatically, typically due to the vanishing or divergence of the coefficients. Recognizing singular points is essential for determining the appropriate method to solve an equation, such as employing the Frobenius method when a regular singular point is present.

Series Solutions (Frobenius Method)

The Frobenius method is a technique used to find series solutions to linear differential equations near a regular singular point. This method expands the solution as a power series with a leading exponent that may be determined by an indicial equation, allowing for the systematic construction of a valid local solution around the singularity.

Transcript

00:01 Okay, so for this program, we first need to rewrite the lde in the following way.

00:10 W double prime plus px, w prime, plus w equals w equals qx times w equals to 0.

00:20 So in this case, px equals to minus 1 over x, and qx equals to minus 1.

00:27 So x equals to 0 is a singular value.

00:32 We can see that the following to limit x times px, when x goes to 0, we have minus 1.

00:41 The limit x goes to 0, x squared times qx is 0.

00:48 That means x equals 0 is a regular singular value with initial equation, r times r minus r minus r equals 0.

00:59 This quadratic equation has two solutions, r equals to 0 and i.

01:04 Equals to 2.

01:05 We choose the larger one to be the exponent.

01:09 So the solution to this ode has an expansion.

01:14 We denote this expansion by w2x.

01:18 So it's a summation k from 0 to infinity 8k, x to k plus 2.

01:23 And we take the derivative for w and the second order derivative.

01:45 So we plug in this expansion to the left side of the ode.

01:50 That gives us the left and side equals to summation k from 0 to infinity, k plus 2 times k plus 1, a sub k x 3 k plus 1, minus summation k from 0 to infinity, k plus 2 times a sub k x to the k plus 1, minus summation k from 0 to infinity, a sub k, x to k plus 3.

02:20 We need to shift the index to merge all these submissions into a single one.

02:27 So first we can merge the first two terms.

02:31 So we have summation k from zero to infinity, k plus two times k times 8k, x to the k plus 1, minus summation k from zero to infinity, 8k, x to the k plus 3.

02:48 So if we consider the first term, if we plug in k equals 0 into the first term, we have 0.

02:54 That means, in fact, this summation should be a start with k from 1.

03:01 Now we can shift the index...

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