Classify singular points of a second order ODE

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Newcastle University Mathematics and Statistics 9 years, 2 months ago

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Question statement

Find and classify all the singular points (excluding the point at infinity) of the equation

$\var{latex(y'')} + \var{latex(y')} + \var{latex(y)} = 0$

Enter your answer for each part as a list of numbers, separated by commas and enclosed in square brackets. For example, if $8$ and $9$ were regular singular points, you would enter [8,9]. If there are no points, enter [].

Give any introductory information the student needs.

			Name	Type	Generated Value

singular_points	list	[ 2, -4 ]
extra_points	list	[ -5, -3 ]
points	list	[ 2, -4, -5, -3 ]
a	list	[ 2, 2, 0, 0 ]
b	list	[ 2, 2, 1, 0 ]
c	list	[ 0, 3, 0, 2 ]
p	list	[ 0, 0, 1, 0 ]
q	list	[ -2, 1, 0, 2 ]
y''	string	\left ( x - 2 \right )^{ 2 } \
y'	string	\left ( x - 2 \right )^{ 2 } \
y	string	\left ( x + 4 \right )^{ 3 } \
is_essential	list	[ false, false ]
essential_points	list	[ ]
regular_points	list	[ 2, -4 ]

			Name	Type	Generated Value

Automatically regenerate variables?

Name

Data type

Value

xxxxxxxxxx
 
shuffle(list(-5..5))[0..random(0,1,1,1,2,2,2,2)]

JME function reference

Description

Some randomly picked points to be singular.

Describe what this variable represents, and list any assumptions made about its value.

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Generated value: `list`

[ 2, -4 ]

→ Used by:

a
essential_points
extra_points
is_essential
points
regular_points
y''

Advice

Parts

Part a) Gap-fill
1. Gap Mathematical expression
  Add
Part b) Gap-fill
1. Gap Mathematical expression
  Add

Gap-fill

Ask the student a question, and give any hints about how they should answer this part.

Regular singular points: Unnamed gap

Advice

First write the equation in the form

$y''+p(x)y'+q(x)y=0,$

so we have

$y'' + \var{latex(coefficient(points,p,'y\''))} + \var{latex(coefficient(points,q,'y'))} = 0$

That is,

$\begin{align} p(x) &= \var{latex(coefficient(points,p,''))}, & q(x) &= \var{latex(coefficient(points,q,''))}. \end{align}$

This equation has no singular points, i.e. all points are analytic.

$x = \var{singular_points[0]}$ is a singular point of the equation. This is a regular essential point if both $(\simplify{x-{singular_points[0]}})p(x)$ and $(\simplify{x-{singular_points[0]}})^2q(x)$ are analytic at $x = \var{singular_points[0]}$ .

So first form

$(\simplify[]{x-{singular_points[0]}})p(x) = \var{latex(coefficient(points,[p[0]+1]+p[1..len(p)],''))}$

This is {if(p[0]+1>=0,"analytic","singular")} at $x = \var{singular_points[0]}$ .

Next, form

$(\simplify[]{x-{singular_points[0]}})^2q(x) = \var{latex(coefficient(points,[q[0]+2]+q[1..len(q)],''))}$

This is {if(q[0]+2>=0,"analytic","singular")} at $x = \var{singular_points[0]}$ .

Hence $x = \var{singular_points[0]}$ is {if(is_essential[0],"an essential","a regular")} singular point.

$x = \var{singular_points[1]}$ is another singular point of the equation, so form

$(\simplify[]{x-{singular_points[1]}})p(x) = \var{latex(coefficient(points,[p[0],p[1]+1]+p[2..len(p)],''))}$

This is {if(p[1]+1>=0,"analytic","singular")} at $x = \var{singular_points[1]}$ .

Next, form

$(\simplify[]{x-{singular_points[1]}})^2q(x) = \var{latex(coefficient(points,[q[0],q[1]+2]+q[2..len(q)],''))}$

This is {if(q[1]+2>=0,"analytic","singular")} at $x = \var{singular_points[1]}$ .

Hence $x = \var{singular_points[1]}$ is {if(is_essential[1],"an essential","a regular")} singular point.

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Classify singular points of a second order ODE

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Christian Lawson-Perfect 5 years, 10 months ago

Christian Lawson-Perfect 5 years, 10 months ago

Newcastle University Mathematics and Statistics 9 years, 2 months ago

Error in variable testing condition

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Generated value: `list`

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Classify singular points of a second order ODE

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

Feedback

History

Comment

Checkpoint description

Christian Lawson-Perfect 5 years, 10 months ago

Christian Lawson-Perfect 5 years, 10 months ago

Newcastle University Mathematics and Statistics 9 years, 2 months ago

Error in variable testing condition

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Warning

Generated value: list

Parts

Gap-fill

Generated value: `list`