Determine all solutions of the differential equation:

dy/dx =(x-y+3)^2

Please give the detailed solution with complete analysis, not alone
the computational part.

PROBLEM

dy/dx =(x-y+3)^2

SOLUTION

Do w=x-y+3, then dw/dx=1-dy/dx, you have that dy/dx=1-dw/dx (1) Now,
replace w and (1) in the original equation

New equation: 1-dw/dx=w^2
Then, dw/dx=1-w^2 (2)

To solve (2)
you need to separate the variables

dw/(1-w^2)=dx  Now youŽll integrate both parts of the equation

1/2 Ln ((1+w)/(1-w))= x + c --> you must review integration tables

Algebra separations

Ln ((1+w)/(1-w)) = 2x + 2c

(1+w)/(1-w)= e^(2x + 2c)

(1+w)/(1-w)= e^2x * e^2c

(1+w)/(1-w)= k*e^2x (3) k is an arbitrary constant and it replaces e^2c

from the beginning you have w=x-y+3

Substitution of w in (3)

(1+x-y+3)/(1-x+y-3)=k*e^2x

Algebra separations, again

(4+x-y)/(-x+y-2)=k*e^2x

4+x-y=k*e^2x(-x+y-2)

4+x-y=-(k*e^2x)x+(k*e^2x)y-2(k*e^2x)

4+x+(k*e^2x)x+2(k*e^2x)=(k*e^2x)y+y

4+x+kx*e^2x+2k*e^2x=(k*e^2x+1)y

Then y = (4+x+kx*e^2x+2k*e^2x)/((k*e^2x+1) is the solution.

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Request for Answer Clarification by dicepaul-ga on 16 Sep 2004 14:36 PDT

Sorry, I needed solutions with  complete analysis , not only the
computational part .
Also  there are Four different solutions of the given differential equation .
(This is a question of graduate level.)

Regards

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Clarification of Answer by cerebro-ga on 17 Sep 2004 12:51 PDT

this differential equation of order 1 can only have one solution
because its form is dy/dx=f(ax+by+c)

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Request for Answer Clarification by dicepaul-ga on 18 Sep 2004 16:57 PDT

sorry this answer is not acceptable.

if possible please refer to the below book.
Reference book: 
Graduate Text in mathematics
Readings in Mathematics
Author: Wolfgang Walter
Publisher: Springer
isbn: 0-387-98459-3

i will try to give a sketch of solution to similar question so that
you can understand the requirement.

i dont know how we can call truce on this, if you know that please let me know.

i am sorry for the trouble.

thanks,

Hi, dicepaul-ga:

Here's one approach.  Let w = x-y+3.

Then dw/dx = 1 - dy/dx =  (what, in terms of w?)

Solve that to relate x to w, and then y = x-w+3 is the general solution.

Some discussion on your part of the possible initial conditions would be in order.

regards, mathtalk-ga

As regards the number of solutions, perhaps the more complete
statement would be that a nonlinear first-order differential equation
satisfying certain continuity/smoothness conditions on the right-hand
side (here (x-y+3)^2) will have a single curve through each point in
the (x,y)-plane (ie. one solution for each initial condition).

This leads to a family of "solution" curves that "cover" the plane. 
Two solution curves can never intersect in the (x,y)-plane (unless
they are identical).

Stated in terms of function w, as cerebro seems to have adopted from
my earlier Comment, the differential equation becomes "autonomous":

  dw/dx = 1 - w^2

meaning that there is no explicit dependence on x.  In this situation
any "translate" of a solution curve w(x) to w(x+a) gives a parallel
curve (or an identical "curve" if the solution w is a constant, as
happens here with w = 1 or w = -1).

I believe it is in the sense that the solutions, when viewed as
families of curves in the (x,y)-plane, fall into these parallel
classes that dicepaul-ga wishes to have Clarified what the
possibilities are.  From the perspective of the solutions for w, we
see that either a solution is always above 1, always below -1, or
always strictly between -1 and +1 (unless it is one of the trivial
solutions +1 or -1).

regards, mathtalk-ga

Greetings, dicepaul!

The answer to your last question can be found near the bottom of the
Google Answers Help page:

http://www.answers.google.com/answers/help.html#followup  

Best wishes,

aceresearcher