Differential equations 2

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Violeta CIT 6 years, 4 months ago

Created this as a copy of Kieran's copy of Julie's copy of First order differential equations 4.

Name	Status	Author	Last Modified
Julie's copy of First order differential equations 4	draft	Julie Crowley	08/10/2016 20:24
Kieran's copy of Julie's copy of First order differential equations 4	draft	Kieran Mulchrone	03/04/2017 11:22
Differential equations 2	draft	Violeta CIT	20/11/2018 16:20
Simon's copy of Julie's copy of First order differential equations 4	draft	Simon Thomas	18/03/2019 13:50

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Question statement

Separate the variables:

Find the solution of:

$\frac{dy}{dx}=\simplify[std]{(1+y^2)/({a}+{b}x)}$
which satisfies

$y(1)=\var{u}$

Note that if $\pi$ is in your expression you enter it as pi.

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			Name	Type	Generated Value

a	integer	6
b	integer	9
u	integer	0
t	integer	1
v	integer	0

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Generated value: `integer`

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1. Gap Mathematical expression
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Solution is:

$y=\;\;$ Unnamed gap

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Advice

We can separate the variables to get

$\frac{dy}{dx}= \simplify[std]{(1+y^2)/({a}+{b}x)} \Rightarrow \frac{1}{1+y^2}\frac{dy}{dx}=\simplify[std]{{a}+{b}x}$

On integrating we get:

$\arctan(y)=\frac{1}{\var{b}}\ln\left(\left|\var{a}+\var{b}x\right|\right)+A \Rightarrow y=\tan\left(\frac{1}{\var{b}}\ln\left(\left|\var{a}+\var{b}x\right|\right)+A\right)$
To fix the arbitrary constant of integration we use the condition

$y(1)=\var{u}$ .

As $\arctan(\var{u})=\var{v}$ we see that $\displaystyle{A = \var{v}-\frac{1}{\var{b}}\ln(|\var{a+b}|)}$ .

Hence the solution is

$y=\simplify[std]{tan(({((t * (t -1)) / 2)} * (pi / 3)) + ({((t -1) * (t -2)) / 2} * (pi / 4)) + ((1 / {b}) * ln(abs(({a} + ({b} * x)) / {a + b}))))}$

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Taxonomy: mathcentre

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Violeta CIT 6 years, 4 months ago

Error in variable testing condition

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Generated value: `integer`

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Differential equations 2

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

History

Comment

Checkpoint description

Violeta CIT 6 years, 4 months ago

Error in variable testing condition

Error

Warning

Generated value: integer

Parts

Gap-fill

Generated value: `integer`