Find one singular points of the following differential equations. Classify them as regular...

Question:

Find one singular points of the following differential equations. Classify them as regular singular points or irregular singular points.

{eq}\;\;\;\;\;\;\;\;\; x^2 (x - 2)^2 y'' + (x - 2) y' + y = 0 {/eq}.

Singular Points of Differential Equations:

Suppose we have a second-order linear homogeneous differential equation

{eq}y''+p_1(x)y'+p_0(x)y=0 \, . {/eq}

A point {eq}x_0 {/eq} is a singular point of this differential equation if either {eq}p_1(x) {/eq} or {eq}p_0(x) {/eq} has a singularity at {eq}x=x_0 {/eq}. A singular point is said to be a regular singular point if {eq}(x-x_0)p_1(x) {/eq} and {eq}(x-x_0)^2p_0(x) {/eq} are continuous or have removable discontinuities at {eq}x {/eq}.

Solutions to differential equations near their regular singular points are not too difficult to classify; they can be "no worse in behavior" than the solution to a Cauchy-Euler equation. That is, they will always behave at worst like {eq}(x-x_0)^a {/eq} or {eq}(x-x_0)^a\ln(x-x_0) {/eq} for some {eq}a {/eq}.

Answer and Explanation:

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To classify the singular points of the differential equation

{eq}x^2(x-2)^2y''+(x-2)y'+y=0 \, , {/eq}

we rewrite it with leading coefficient 1....

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First-Order Linear Differential Equations

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Chapter 16 / Lesson 3
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In this lesson you'll learn how to solve a first-order linear differential equation. We first define what such an equation is, and then we give the algorithm for solving one of that form. Specific examples follow the more general description of the method.


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