Determine the singular points of the given differential equation. Classify each singular point as regular or irregular. (Enter your answers as a comma-separated list. Include both real and complex singular points. If there are no singular points in a certain category, enter NONE.) x3y'' + 7x2y' + 4y = 0

x³y'' + 7x²y' + 4y = 0

has irregular singular point at x = 0.

Step-by-step explanation:

Consider the differential equation

y'' + P(x)y' + Q(x)y = 0 (1)

A point x = x_0 is called an ORDINARY POINT of (1) if both functions P(x) and Q(x) are analytic (differentiable) at x = x_0.

If the point x = x_0 is not an ordinary point, then it is a SINGULAR POINT.

There are two types of singular points:

REGULAR SINGULAR POINTS

IRREGULAR SINGULAR POINTS

A singular point x = x_0 of (1) is called REGULAR if both (x - x_0)P(x) and (x - x_0)²Q(x) are analytic at x = x_0.

otherwise, the singular point is called IRREGULAR.

EXAMPLE:

Determine if x = 0 is an ordinary point or singular point for the differential equation

x³y'' + 7x²y' + 4y = 0 (2)

First, we rewrite (2) to be in form of (1) by dividing through by x³

y'' + (7/x)y' + (4/x³)y = 0

Comparing with (1)

P(x) = 7/x

Q(x) = 4/x³

At x = 0

P(x) = 7/0 = infinity

Q(x) = 4/0 = infinity

Both P(x) and Q(x) are nonanalytic, so the point x = 0 is not an ordinary point.

Again

(x - 0)P(x) = 7

(x - 0)Q(x) = 4/x

At x = 0

(x - 0)P(x) = 7

(x - 0)Q(x) = infinity

Only (x - 0)P(x) is analytic, so x = 0 is an IRREGULAR SINGULAR POINT.

To determine the singular point of a differential equation, all we need to do is find the point that satisfies the properties of a singular point as explained above.

x³y'' + 7x²y' + 4y = 0

has irregular singular point at x = 0.

why do wages differ across industries

step-by-step explanation :

micro economists are concerned with economics on an industry or individual firm basis.

macro economists are concerned with national and international level

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step-by-step explanation: