Julie's copy of First order differential equations 5

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Julie Crowley

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Julie Crowley 8 years, 5 months ago

Published this.

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Julie Crowley 11 years, 1 month ago

Created this as a copy of First order differential equations 5.

Name	Status	Author	Last Modified
Julie's copy of First order differential equations 5	draft	Julie Crowley	08/10/2016 20:25
Simon's copy of Julie's copy of First order differential equations 5	draft	Simon Thomas	20/03/2019 09:09
Maria's copy of Simon's copy of Julie's copy of First order differential equations 5	draft	Maria Aneiros	27/05/2019 09:04
Integrating Factor 1	Ready to use	Nick McCullen	12/10/2020 17:27
Integrating Factor 2	Ready to use	Nick McCullen	06/10/2020 11:21
1st order ODE - Initial Conditions and new value	Ready to use	Nick McCullen	09/11/2020 22:02

There are 39 other versions that do you not have access to.

Question statement

Integrating Factor Method:

Find the solution of:

$x\frac{dy}{dx}+\var{a}y=\simplify[std]{{b}x^{n}}$

which satisfies $\displaystyle{y(1)=\simplify[std]{{b*(c+1)}/{a+n}}}$

Give any introductory information the student needs.

			Name	Type	Generated Value

a	integer	6
c	integer	2
b	integer	3
n	integer	5

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Generated value: `integer`

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"Unnamed part" → "Unnamed gap" - Correct answer

Parts

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1. Gap Mathematical expression
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Solution is:

$y=\;\;$ Unnamed gap

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Advice

$\displaystyle{x\frac{dy}{dx}+\var{a}y=\simplify[std]{{b}x^{n}}}$ is a linear equation of the special type where we can multiply both sides by $\simplify[std]{x^{a-1}}$ to get:

$\frac{d(x^{\var{a}}y)}{dx}=\simplify[std]{{b}x^{a+n-1}}$
We can integrate both sides to get:

$x^{\var{a}}y=\simplify[std]{{b}/{a+n}x^{a+n}+A}$
to determine

$A$ we use the condition

$\displaystyle{y(1)=\simplify[std]{{b*(c+1)}/{a+n}}}$ and we see that:

$A = \simplify[std]{{b*(c+1)}/{a+n} - {b}/{a+n} = {b*c}/{a+n}}$
and so the solution is:

$x^{\var{a}}y=\simplify[std]{{b}/{a+n}x^{a+n}+{b*c}/{a+n}} \Rightarrow y=\simplify[std]{{b}/{a+n}*(x^{n}+{c}*x^{-a})}$

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Julie's copy of First order differential equations 5

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Taxonomy: mathcentre

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Taxonomy: Context

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Checkpoint description

Julie Crowley 8 years, 5 months ago

Julie Crowley 11 years, 1 month ago

Error in variable testing condition

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Warning

Generated value: `integer`

Parts

Gap-fill

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Error

Julie's copy of First order differential equations 5

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

History

Comment

Checkpoint description

Julie Crowley 8 years, 5 months ago

Julie Crowley 11 years, 1 month ago

Error in variable testing condition

Error

Warning

Generated value: integer

Parts

Gap-fill

Generated value: `integer`