Harry's copy of Separable first order ODE with boundary condition

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Harry Flynn

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Harry Flynn 7 years, 1 month ago

Created this as a copy of Separable first order ODE with boundary condition.

Name	Status	Author	Last Modified
Separable first order ODE with boundary condition	Ready to use	Newcastle University Mathematics and Statistics	20/11/2019 14:50
Harry's copy of Separable first order ODE with boundary condition	draft	Harry Flynn	09/03/2018 14:22

Question statement

The thickness of ice on water, $x(t)$ , grows according to the equation:

$\frac{dx}{dt}=\simplify[std]{{a}/(x+{b})^{n}}$
Given that

$x(0)=0$ find

$x(t)$ .

Give any introductory information the student needs.

			Name	Type	Generated Value

a	integer	5
b	integer	8
n	integer	3

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Value

xxxxxxxxxx
 
random(1..9)

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Description

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

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Generated value: `integer`

This variable doesn't seem to be used anywhere.

Parts

Gap-fill
1. Gap Mathematical expression
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Gap-fill

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The thickness at time $t$ is given by:

$x(t)=\;\;$ Unnamed gap

Input all numbers as integers or fractions – not as decimals.

Advice

On rearranging the equation we get $\displaystyle{\simplify[std]{(x+{b})^{n}*(dx/dt) = {a}}}$ and on integrating we obtain:
$\displaystyle{\simplify[std]{(x+{b})^{n+1}/{n+1}={a}t +A} \Rightarrow x+\var{b}=(A+\var{a*(n+1)}t)^{1/\var{n+1}}}$

Using the condition $x(0)=0$ gives $\displaystyle{A^{1/\var{n+1}}=\var{b} \Rightarrow A=\var{b^(n+1)}}$

Hence the solution is:

$x(t) = \simplify[std]{({(b ^ (n + 1))} + ({(a * (n + 1))} * t)) ^ (1 / {(n + 1)}) - {b}}$

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Harry's copy of Separable first order ODE with boundary condition

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Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

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Checkpoint description

Harry Flynn 7 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

Harry's copy of Separable first order ODE with boundary condition

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

History

Comment

Checkpoint description

Harry Flynn 7 years, 1 month ago

Error in variable testing condition

Error

Warning

Generated value: integer

Parts

Gap-fill

Generated value: `integer`