Solve a separable first order ODE,

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Newcastle University Mathematics and Statistics 9 years, 2 months ago

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Name	Status	Author	Last Modified
Solve a separable first order ODE,	Ready to use	Newcastle University Mathematics and Statistics	20/11/2019 14:50
Simon's copy of Solve a separable first order ODE,	draft	Simon Thomas	20/03/2019 09:39
Grainne's copy of Solve a separable first order ODE,	draft	Grainne Read	14/05/2019 15:02
Maria's copy of Solve a separable first order ODE,	draft	Maria Aneiros	27/05/2019 09:04

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Question statement

You are given the differential equation

$\simplify{{a1}*x*y*y'}=\var{b1}+y^2,$

satisfying $y(\var{c1})=\var{d1}$ .

The solution can be written in the form $y=\pm\sqrt{f(x)}$ , where $f(x)$ is some function of $x$ .

Give any introductory information the student needs.

			Name	Type	Generated Value

a1	integer	4
c1	decimal	dec("4")
b1	number	-6
d1	number	6

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The differential equation is separable, and we can therefore write

$\int{\!\frac{y}{\var{b1}+y^2}\,\mathrm{d}y}=\frac{1}{\var{a1}}\int{\!\frac{1}{x}\,\mathrm{d}x},$

which can be integrated to give

$\frac{1}{2}\ln\lvert\var{b1}+y^2\rvert=\frac{1}{\var{a1}}\ln\lvert x\rvert+c.$

Exponentiating both sides leads to

$\sqrt{\var{b1}+y^2}=\simplify{Ax^(1/{a1})}$

and, on rearranging for $y$ (and redefining $A$ ), we have

$y=\pm\sqrt{\simplify{A*x^(2/{a1})-{b1}}}.$

Then we have

$\var{d1}=y(\var{c1})=\pm\sqrt{\simplify{A*{c1}^(2/{a1})-{b1}}},$

$A=\simplify[std]{({d1}^2+{b1})/{c1}^(2/{a1})}=\simplify{{d1^2+b1}/{c1^(2/a1)}}.$

Then the full solution is

$y=\pm\sqrt{\simplify{{d1^2+b1}/{c1^(2/a1)}*x^(2/{a1})-{b1}}}.$

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Bill Foster 6 years, 1 month ago

Newcastle University Mathematics and Statistics 9 years, 2 months ago

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Solve a separable first order ODE,

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

Feedback

History

Comment

Checkpoint description

Bill Foster 6 years, 1 month ago

Newcastle University Mathematics and Statistics 9 years, 2 months ago

Error in variable testing condition

Error

Warning

Generated value: integer

Parts

Gap-fill

Generated value: `integer`