Simon's copy of Julie's copy of Second order differential equations 2

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Simon Thomas 6 years ago

Created this as a copy of Julie's copy of Second order differential equations 2.

Name	Status	Author	Last Modified
Julie's copy of Second order differential equations 2	draft	Julie Crowley	03/02/2021 13:59
Simon's copy of Julie's copy of Second order differential equations 2	draft	Simon Thomas	03/02/2021 13:59

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Question statement

Method of undetermined coefficients:

Find the solution of:
\[\simplify[std]{(d^2y/dx^2)+{2*a}*(dy/dx)+{a^2}y}=0\]
which satisfies $y(0)=\var{c}$ and $y(1)=\var{d}$.

Give any introductory information the student needs.

			Name	Type	Generated Value

a	integer	-7
f1	number	-5.995
c	integer	6
d	integer	5
g	number	-6.0058381985
f	number	-5.9954405902
m	number	347.5309889707
s	integer	-1
ytick	integer	35

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Name

Data type

Value

xxxxxxxxxx
 
s*random(1..7)

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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`

-7

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Parts

Gap-fill
1. Gap Mathematical expression
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Gap-fill

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{graphdisplay()}

Solution is:

$y=\;\;$Unnamed gap

Input all numbers correct to 3 decimal places.

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Note that the points $(0,\var{c}),\;(1,\var{d})$ which the curve has to go through are marked on the graph.

Advice

The auxillary equation is $\simplify[std]{lambda^2+{2*a}lambda+{a^2}}=0$.

On solving this equation we get $\lambda=\var{-a}$ twice.

Hence the general solution is:
\[y = \simplify[std]{A*e^({-a}x)+B*x*e^({-a}x)}\]
The boundary conditions give:

$y(0)=\var{c} \Rightarrow A=\var{c}$

$y(1)=\var{d} \Rightarrow \simplify[std]{Ae^{-a}+Be^{-a}={d}}\Rightarrow A+B = \simplify[std1]{{d}e^{a}}$

So $B=\simplify[std1]{{d}e^{a}-{c}}=\var{f1}$ to 3 decimal places.

Hence the solution is:
\[y=\simplify[std1]{(({c} * Exp(({( - a)} * x))) + ({f1} * x * Exp(({( - a)} * x))))}\]

{advicedisplay()}

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This question is used in the following exam:

Differential Equations (Multiple types) by Simon Thomas in Differential Equations.

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Simon's copy of Julie's copy of Second order differential equations 2

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Taxonomy: mathcentre

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Simon Thomas 6 years ago

Error in variable testing condition

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Generated value: `integer`

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Gap-fill

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Simon's copy of Julie's copy of Second order differential equations 2

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

History

Comment

Checkpoint description

Simon Thomas 6 years ago

Error in variable testing condition

Error

Warning

Generated value: integer

Parts

Gap-fill

Generated value: `integer`