anonymous
  • anonymous
Solve the following D.E., \[\frac{dy}{1+y^2}=\frac{dx}{1+x^2}\], obtaining the result in algebraic form
Mathematics
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SOLVED
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chestercat
  • chestercat
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anonymous
  • anonymous
the solution given by the text is \[y=x+C(1+xy)\]
lgbasallote
  • lgbasallote
i believe you should cross multiply first \[(1 + x^2)dy - (1+y^2)dx = 0\]
myininaya
  • myininaya
You just integrate both sides of that equation you have @ebbflo

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myininaya
  • myininaya
You already have the variables separated
UnkleRhaukus
  • UnkleRhaukus
\[\int \frac{1}{1+z^2}\text dz=\arctan z+c\]
myininaya
  • myininaya
Is your trouble writing it as an algebraic expression?
anonymous
  • anonymous
I see all of your points and have tried those methods but do not obtain the given answer....
myininaya
  • myininaya
If so.. Don't forget you only need one constant (and just put it on one side of your equation) Remember the inverse of tan inverse which is tan lol also remember the formula tan(a+b)=(tan(a)+tan(b))/(1-tan(a)*tan(b)) Then remember that tan(constant) is still a constant)
myininaya
  • myininaya
Remember @ebbflo it does say write as an algebraic expression.
myininaya
  • myininaya
You get trigonometric expression right after integration
myininaya
  • myininaya
You want to transform that to an algebraic one using the hint I just gave you
myininaya
  • myininaya
Also recall tan(arctan(p))=p
myininaya
  • myininaya
You will use that too
anonymous
  • anonymous
thanks, I understand all the points you have made
myininaya
  • myininaya
What about the solution? Have you got the desired solution?
myininaya
  • myininaya
Or do you need more help?
anonymous
  • anonymous
no, got it, i had the solution, just don't quite understand "why" the book's solution is written in the form it is... the first copyright is 1943, maybe its a style thing...
anonymous
  • anonymous
@myininaya , you can use Latex so that your math text is more clear
myininaya
  • myininaya
That one formula is a trig formula The expansion for tan(a+b) let me know if you need anything else
anonymous
  • anonymous
as i said, I had an equivalent solution, I was just curious as to why the book chose to write the solution in the form it did...
myininaya
  • myininaya
Was your form algebraic?
myininaya
  • myininaya
Can you write what you have?
anonymous
  • anonymous
I have the solution given, I think the book just wanted a form with one arbitrary constant, i generally don't like my solutions to have "y" on the LHS and RHS when it is not absolutely necessary...it was really ore of a "style" question, sorry I should have specified
myininaya
  • myininaya
If the equation can be written in the form the book gives, then your answer is right I got the same answer your book got I wrote it in a different form but it is still correct
anonymous
  • anonymous
yes I understand, thank you...as I said I was questioning the style, and now that I think of it it was probably that the books form only take a single line of text whereas mine did not...;)
UnkleRhaukus
  • UnkleRhaukus
differential equations will often have \(y\) on both sides of the solution, but that is alright. we were only trying to get rid of derivatives replace them with constants
anonymous
  • anonymous
@unkle agreed, but it is my preference not to do so when not necessary
anonymous
  • anonymous
I should have been more clear, I was not so much looking for "help" with the problem, but a different perspective on how another might write the solution
UnkleRhaukus
  • UnkleRhaukus
these are some solution to differential equations ,
anonymous
  • anonymous
in order to make a decision as to what a student may find more clear
UnkleRhaukus
  • UnkleRhaukus
your solution must include as many arbitrary constants as the order of the differential equation
anonymous
  • anonymous
exactly uncle, in those solution you provided, the "y" cannot be written explicitly in terms of "x", at least not in a single expression
UnkleRhaukus
  • UnkleRhaukus
well, it can but i just looks awful , these are 'neater'
UnkleRhaukus
  • UnkleRhaukus
looking at (e)
anonymous
  • anonymous
yeah that one can but what about (c)?
anonymous
  • anonymous
(e) more or less is already, not much more to do on that one...
UnkleRhaukus
  • UnkleRhaukus
i think it is clearer to write (c) like \[x = y^2(\ln y + c)\] i dont know why they chose the form they did
UnkleRhaukus
  • UnkleRhaukus
implicit solutions are fine for DE's

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