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Old 2007-07-28, 22:19   #1
ShiningArcanine
 
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Default Differential equation question

Has a deterministic solution to the differential equation y' = y + x been found or is it still unsolved? Also, is there any application where solving that particular differential equation is useful?
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Old 2007-07-29, 02:49   #2
wpolly
 
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y(x)=C exp(x)-x-1
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Old 2007-07-29, 09:04   #3
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What is C in the context of an initial value problem?

Last fiddled with by ShiningArcanine on 2007-07-29 at 09:23 Reason: Simplifying question
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Old 2007-07-29, 10:15   #4
VolMike
 
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Quote:
Originally Posted by ShiningArcanine View Post
What is C in the context of an initial value problem?
C is some complex (in general case) value
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Old 2007-07-29, 10:41   #5
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So is it possible to plot the solution of y as a function of x using this or must it be approximated?

Edit: Also, I am curious, how was the solution that wpolly posted found? Was it found by guessing and proven by substitution?

Last fiddled with by ShiningArcanine on 2007-07-29 at 10:56
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Old 2007-07-29, 11:35   #6
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Quote:
Originally Posted by ShiningArcanine View Post
So is it possible to plot the solution of y as a function of x using this or must it be approximated?

Edit: Also, I am curious, how was the solution that wpolly posted found? Was it found by guessing and proven by substitution?

Generally, there is a family of solutions (infinite numbers): any solution can be plotted by substituing C for some real value. Maybe you have some additional information for your problem: f.e. statement like y(a)=b. If so, you can calculate single C value and then plot the solution,otherwise there are infinite solutions (one solution for each C value) to be plotted.

There are many methods for symbolical solving different types of DE. Some of them described at http://eqworld.ipmnet.ru/ru/solutions/ode/ode-toc1.htm (russian headers, but english explanation). Your's linear DE can be solved by method,described at http://eqworld.ipmnet.ru/en/solutions/ode/ode0103.pdf You can aslo use (if have) CAS Mathematica or Maple to solve DE.
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Old 2007-07-29, 11:47   #7
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Is it known how to solve y(x) for y' = x + y if the vector (0,1) is a solution of y(x)?

Edit: Also, as I asked in my original post, is there an application where solving y' = x + y is useful?

Last fiddled with by ShiningArcanine on 2007-07-29 at 11:51
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Old 2007-07-29, 12:15   #8
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Quote:
Originally Posted by ShiningArcanine View Post
Is it known how to solve y(x) for y' = x + y if the vector (0,1) is a solution of y(x)?

Edit: Also, as I asked in my original post, is there an application where solving y' = x + y is useful?
It seems I undrestand you; vector solution (0,1) of y(x) is the same with y(0)=1. If so, you just need to solve linear equation respectively to C. I.e. you have common solution y(x)=C* exp(x)-x-1 of y'=x+y (it must be found before, by CAS or manually)

If y(0)=1 then (by substituting x with 0 and y with 1) you'll have:
1=C*exp(0)-0-1 or C=2
Thus, you have y(x)=-1-x+2*exp(x) as a solution which can be plotted.

If I still didn't get you, give me the full description of your problem

P.S. I don't know the phisical or math application for which this equation can be usefull
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Old 2007-07-29, 12:52   #9
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Thanks. That answers my question.
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