Separable variables (1)

Error

There was an error loading the page.

Contributors

Stephen Bowlzer

History

○

Stephen Bowlzer 8 years, 9 months ago

Created this as a copy of Solve a separable first order ODE, .

Name	Status	Author	Last Modified
Solve a separable first order ODE,	Ready to use	Newcastle University Mathematics and Statistics	20/11/2019 14:50
Separable variables (1)	draft	Stephen Bowlzer	21/06/2016 16:22
Separable variables 1 violeta	draft	Violeta CIT	23/11/2018 08:51
Aoife's copy of Separable variables 1 violeta	draft	Aoife O'Brien	07/03/2019 11:05
Simon's copy of Solve a separable first order ODE,	draft	Simon Thomas	20/03/2019 09:58
Maria's copy of Separable variables (1)	draft	Maria Aneiros	27/05/2019 09:04
Maria's copy of Solve a separable first order ODE,	draft	Maria Aneiros	27/05/2019 09:04

There are 21 other versions that do you not have access to.

Question statement

Equations which can be written in the form

$\dfrac{\mathrm{d}y}{\mathrm{d}x} = f(x), \dfrac{\mathrm{d}y}{\mathrm{d}x} = f(y), \dfrac{\mathrm{d}y}{\mathrm{d}x} = f(x)f(y)$

can all be solved by integration.

In each case it is possible to separate the $x$ 's to one side of the equation and the $y$ 's to the other

Solving such equations is therefore known as solution by separation of variables

Question

Find the solution of the differential equation

$(\var{a1}+x)\dfrac{\mathrm{d}y}{\mathrm{d}x}=\var{b1}+y$

satisfying the condition that $y = \var{d1}$ when $x = \var{c1}$

Give any introductory information the student needs.

			Name	Type	Generated Value

a1	number	-4
c1	number	1
b1	number	1
d1	number	9

Automatically regenerate variables?

Name

Data type

Value

xxxxxxxxxx
 
random(1..9)*sign(random(-1,1))

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: `number`

-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.

$y=$ Unnamed gap (Do not enter decimals in your answer.)

Advice

The differential equation is separable, so we can write

$\int{\!\frac{1}{\var{b1}+y}\,\mathrm{d}y} = \int{\!\frac{1}{\var{a1}+x}\,\mathrm{d}x},$

then

$\ln\lvert\var{b1}+y\rvert=\ln\lvert\var{a1}+x\rvert+c,$

$y=\simplify{A({a1}+x)-{b1}},$

which is the general solution of the equation.

Now,

$\var{d1}=y(\var{c1})=\simplify[std]{A({a1}+{c1})-{b1}},$

$A=\simplify[std]{({d1}+{b1})/({a1}+{c1})}=\simplify{{d1+b1}/{a1+c1}},$

and then the full solution is

$y=\simplify[std]{{d1+b1}/{a1+c1}({a1}+x)-{b1}}=\simplify{{(d1*a1-b1*c1)}/{a1+c1}+{(d1+b1)}*x/{a1+c1}}.$

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:

Differential Equations (1) by Stephen Bowlzer in Stephen's workspace.
Separable ODEs by David Graham in David's workspace.

Loading...

Error

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

History

Comment

Checkpoint description

Stephen Bowlzer 8 years, 9 months ago

Error in variable testing condition

Error

Warning

Generated value: `number`

Parts

Gap-fill

Loading...

Error

Separable variables (1)

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

History

Comment

Checkpoint description

Stephen Bowlzer 8 years, 9 months ago

Error in variable testing condition

Error

Warning

Generated value: number

Parts

Gap-fill

Generated value: `number`