Julie's copy of Second order differential equations 1

Error

There was an error loading the page.

Contributors

Julie Crowley

History

○

Julie Crowley 8 years, 5 months ago

Published this.

○

Julie Crowley 11 years, 1 month ago

Created this as a copy of Second order differential equations 1.

Name	Status	Author	Last Modified
Julie's copy of Second order differential equations 1	draft	Julie Crowley	03/02/2021 13:59
Simon's copy of Julie's copy of Second order differential equations 1	draft	Simon Thomas	03/02/2021 13:59

There are 10 other versions that do you not have access to.

Question statement

Method of undertermined coefficients

Solve:
\[\simplify[std]{(d^2y/dx^2)+{2*a}*(dy/dx)+{a^2+b^2}y}=0\]
which satisfies $y(0)=1$ and $y'(0)=\var{c}$ (where prime denotes the derivative).

Give any introductory information the student needs.

			Name	Type	Generated Value

a	integer	-4
c	integer	3
b	integer	7

Automatically regenerate variables?

Name

Data type

Value

xxxxxxxxxx
 
random(-5..5 except 0)

JME function reference

Description

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

Can an exam override the value of this variable?

Generated value: `integer`

-4

This variable doesn't seem to be used anywhere.

Parts

Gap-fill
1. Gap Mathematical expression
  Add

Gap-fill

Ask the student a question, and give any hints about how they should answer this part.

Solution is:

$y=\;\;$Unnamed gap

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

Advice

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

On solving this equation we get $\lambda=\simplify[std]{{-a}+{b}i}$ and $\lambda=\simplify[std]{{-a}-{b}i}$.

Hence the general solution is:
\[y = \simplify[std]{e^({-a}x)(A*sin({b}x)+B*cos({b}x))}\]
Note that
\[y'(x)=\simplify[std]{-{a}e^({-a}x)(A*sin({b}x)+B*cos({b}x))+e^({-a}x)({b}*A*cos({b}x)-{b}*B*sin({b}x))}\]
Using the conditions $y(0)=1$ and $y'(0)=\var{c}$ gives:
\[\begin{eqnarray*} B &=& 1\\ \simplify[std]{{b}A+{-a}B}&=& \var{c} \end{eqnarray*} \]
This gives $\displaystyle{A = \simplify[std]{{c+a}/{b}}}$.

Hence the solution is:

\[y=\simplify[std]{exp({- a} * x) * (cos({b} * x) + ({c+a} / {b}) * sin({b} * x))}\]

Give a worked solution to the whole question.

Use this tab to check that this question works as expected.

There was an error which means the tests can't run:

Part	Test	Passed?
Gap-fill
Gap-fill		Hasn't run yet
Mathematical expression
Mathematical expression		Hasn't run yet

This question is used in the following exams:

Loading...

Error

Julie's copy of Second order differential equations 1

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

History

Comment

Checkpoint description

Julie Crowley 8 years, 5 months ago

Julie Crowley 11 years, 1 month ago

Error in variable testing condition

Error

Warning

Generated value: `integer`

Parts

Gap-fill

Loading...

Error

Julie's copy of Second order differential equations 1

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

History

Comment

Checkpoint description

Julie Crowley 8 years, 5 months ago

Julie Crowley 11 years, 1 month ago

Error in variable testing condition

Error

Warning

Generated value: integer

Parts

Gap-fill

Generated value: `integer`