Integrating Factor 1

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Nick McCullen 4 years, 6 months ago

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Nick McCullen 5 years, 5 months ago

Created this as a copy of Maria's copy of Simon's copy of Julie's 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

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Question statement

Integrating Factor Method:

Find the General Solution of:

$\simplify{{A}x^{{m}}}\simplify{(d y)/(d x)+{B}}x^\simplify{{n}+{m}}y=\simplify{{D}x}^\simplify{{n}+{m}}$

Enter your answer as a formula: use the syntax c*e^(n*x^p) for

$ce^{nx^p}$ , not forgetting the * for multiplying arbitrary constants. Use lowercase $c$ for any constant of integration.

Give any introductory information the student needs.

			Name	Type	Generated Value

G	integer	2
H	integer	1
A	integer	3
n	integer	2
B	integer	9
D	integer	18
m	integer	1
c	integer	1

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Generated value: `integer`

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

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1. Gap Mathematical expression
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$y=\;\;$ Unnamed gap

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Advice

$\simplify{{A}x^{{m}}}\simplify{(d y)/(d x)+{B}}x^\simplify{{n}+{m}}y=\simplify{{D}x}^\simplify{{n}+{m}}$ can be solved by the integrating factor method.

Firstly divide both sides by $\simplify{{A}x^{{m}}}$ to obtain $\simplify{(d y)/(d x)+{B}/{A}}\simplify{x^{n}}y=\simplify{{D}/{A}*x^{n}}$

This is now in the standard form $\displaystyle{\frac{dy}{dx}+f(x)y=r(x)}$ which can be solved using the integrating factor $I=e^{\int f(x)dx}$

In this example, $f(x) =\simplify{{B}/{A} x^{n}}$ so $e^{\int f(x)dx}=e^{\int \simplify{{B}/{A} x^{n} d x}}=e^{\simplify{{B}/{(n+1)*A} x^{n+1}}}$

The integrating factor formula $y = \frac{1}{I}\int r(x) I dx$ can be used with $r(x) = \simplify{{D}/{A}x^{n}}$ to obtain:

$\displaystyle y =e^{\simplify{-{B}/{(n+1)*A} x^{n+1}}} \int \simplify{{D}/{A}x^{{n}}}e^{\simplify{{B}/{({n+1})*A} x^{n+1}}} dx$

This can be solved using the substitution: $u= \simplify{{B}/{(n+1)*A} x^{n+1}}$ so $\frac{du}{dx} = \simplify{{B}/{A} x^{n}}$ and therefore $\simplify{x^{n}} dx = \simplify{{A}/{B}}du$ , which gives:

$\displaystyle y = e^{-u} \int \simplify{{D}/{B}e^u} du = e^{-u} \left[\simplify{{D}/{B} e^u} + c\right] = \simplify{{D}/{B}} + ce^{-u}$

Substituting back $u= \simplify{{B}/{(n+1)*A} x^{n+1}}$ results in a general solution: $y = \simplify{{D}/{B}} + ce^{\simplify{-{B}/{(n+1)*A} x^{n+1}}}$

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Nick McCullen 4 years, 6 months ago

Andrew Barnes was given access to the Integrating Factor 1 4 years, 6 months ago

Nick McCullen 5 years, 5 months ago

Nick McCullen 5 years, 5 months ago

Error in variable testing condition

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Generated value: `integer`

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Integrating Factor 1

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

Feedback

History

Comment

Checkpoint description

Nick McCullen 4 years, 6 months ago

Andrew Barnes was given access to the Integrating Factor 1 4 years, 6 months ago

Nick McCullen 5 years, 5 months ago

Nick McCullen 5 years, 5 months ago

Error in variable testing condition

Error

Warning

Generated value: integer

Parts

Gap-fill

Generated value: `integer`