find particular soln to dy/dx=3-y/x using analytical method

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BRAIN
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find particular soln to dy/dx=3-y/x using analytical method

Postby BRAIN » Sat Apr 25, 2009 9:42 pm

How to find particular solution using analytical method?
if dy/dx=3-y/x, and there is an initial condition that y=2 when x=1

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stapel_eliz
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Postby stapel_eliz » Sat Apr 25, 2009 9:51 pm

BRAIN wrote:How to find particular solution using analytical method?
if dy/dx=3-y/x....

Does the above mean this?

. . . . .

Or this?

. . . . .

What does your book mean by "the analytical method"? How far have you gotten in applying this method?

Please be complete. Thank you! :D

BRAIN
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Re: find particular soln to dy/dx=3-y/x using analytical method

Postby BRAIN » Sun Apr 26, 2009 10:43 am

it is :
. . . . .

I think Analytical solutions means relatively few different equations can be solved analytically, gives exact solution.. maybe question meant that.
Also I have another question and post here that using the same equation and finding numerical solution using euler method, maybe saying use and analytical method meant to seperate this two questions!

I think I need to use dy/dx + Py = Q but I dont know how to deal with this equation. :oops:
Hope this is clear to help.

Thank you.

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Martingale
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Re: find particular soln to dy/dx=3-y/x using analytical method

Postby Martingale » Sun Apr 26, 2009 4:16 pm

BRAIN wrote:it is :
. . . . .

I think Analytical solutions means relatively few different equations can be solved analytically, gives exact solution.. maybe question meant that.
Also I have another question and post here that using the same equation and finding numerical solution using euler method, maybe saying use and analytical method meant to seperate this two questions!

I think I need to use dy/dx + Py = Q but I dont know how to deal with this equation. :oops:
Hope this is clear to help.

Thank you.


Use an Integrating Factor.

FrozenBlood
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Re: find particular soln to dy/dx=3-y/x using analytical method

Postby FrozenBlood » Mon Apr 27, 2009 5:49 pm

Is this for a calculus class or a differential equations class?

Anyway, is a pretty straightforward linear first-order ODE.

First, rewrite it in standard form:

In standard form, is .

Now get the integrating factor :





(let C = 1)

Now multiply x through both sides of :


Rewrite the left side as:

Multiply both sides by dx:



Now take the integral of both sides:





And that's the general solution. But this is an IVP, so you need to solve for C using the initial condition y(1) = 2:






So the particular solution to the IVP is


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