Integrating Factor 2

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Created this as a copy of Integrating Factor 1.

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 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

D	integer	3
a	integer	2
c	integer	4
b	integer	18
n	integer	4

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$\displaystyle{x\frac{dy}{dx}+\var{a}y=\simplify[std]{{b}x^{n}}}$ can be solved by the integrating factor method.

Firstly divide both sides by $x$ to obtain $\displaystyle{\frac{dy}{dx}+\frac{\var{a}}{x}y=\simplify[std]{{b}x^{n-1}}}$

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, $f(x) = \frac{\var{a}}{x}$ so $I=e^{\int f(x)dx}=e^{\int{\frac{\var{a}}{x}dx}}=e^{\var{a}\ln(x)}=x^\var{a}$

Using the integrating factor formula $y = \frac{1}{I} \int r(x) I dx$ with $r(x) =\simplify[std]{{b}x^{n-1}}$ gives a solution:

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

$c$ use the condition

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

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

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

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

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Nick McCullen 4 years, 6 months ago

Andrew Barnes was given access to the Integrating Factor 2 4 years, 6 months ago

Nick McCullen 5 years, 5 months ago

Nick McCullen 5 years, 5 months ago

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Integrating Factor 2

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Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

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Nick McCullen 4 years, 6 months ago

Andrew Barnes was given access to the Integrating Factor 2 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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Generated value: `integer`