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Solve the following D.E.,
\[\frac{dy}{1+y^2}=\frac{dx}{1+x^2}\], obtaining the result in algebraic form
 one year ago
 one year ago
Solve the following D.E., \[\frac{dy}{1+y^2}=\frac{dx}{1+x^2}\], obtaining the result in algebraic form
 one year ago
 one year ago

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ebbfloBest ResponseYou've already chosen the best response.0
the solution given by the text is \[y=x+C(1+xy)\]
 one year ago

lgbasalloteBest ResponseYou've already chosen the best response.0
i believe you should cross multiply first \[(1 + x^2)dy  (1+y^2)dx = 0\]
 one year ago

myininayaBest ResponseYou've already chosen the best response.1
You just integrate both sides of that equation you have @ebbflo
 one year ago

myininayaBest ResponseYou've already chosen the best response.1
You already have the variables separated
 one year ago

UnkleRhaukusBest ResponseYou've already chosen the best response.1
\[\int \frac{1}{1+z^2}\text dz=\arctan z+c\]
 one year ago

myininayaBest ResponseYou've already chosen the best response.1
Is your trouble writing it as an algebraic expression?
 one year ago

ebbfloBest ResponseYou've already chosen the best response.0
I see all of your points and have tried those methods but do not obtain the given answer....
 one year ago

myininayaBest ResponseYou've already chosen the best response.1
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))/(1tan(a)*tan(b)) Then remember that tan(constant) is still a constant)
 one year ago

myininayaBest ResponseYou've already chosen the best response.1
Remember @ebbflo it does say write as an algebraic expression.
 one year ago

myininayaBest ResponseYou've already chosen the best response.1
You get trigonometric expression right after integration
 one year ago

myininayaBest ResponseYou've already chosen the best response.1
You want to transform that to an algebraic one using the hint I just gave you
 one year ago

myininayaBest ResponseYou've already chosen the best response.1
Also recall tan(arctan(p))=p
 one year ago

myininayaBest ResponseYou've already chosen the best response.1
You will use that too
 one year ago

ebbfloBest ResponseYou've already chosen the best response.0
thanks, I understand all the points you have made
 one year ago

myininayaBest ResponseYou've already chosen the best response.1
What about the solution? Have you got the desired solution?
 one year ago

myininayaBest ResponseYou've already chosen the best response.1
Or do you need more help?
 one year ago

ebbfloBest ResponseYou've already chosen the best response.0
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...
 one year ago

ebbfloBest ResponseYou've already chosen the best response.0
@myininaya , you can use Latex so that your math text is more clear
 one year ago

myininayaBest ResponseYou've already chosen the best response.1
That one formula is a trig formula The expansion for tan(a+b) let me know if you need anything else
 one year ago

ebbfloBest ResponseYou've already chosen the best response.0
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...
 one year ago

myininayaBest ResponseYou've already chosen the best response.1
Was your form algebraic?
 one year ago

myininayaBest ResponseYou've already chosen the best response.1
Can you write what you have?
 one year ago

ebbfloBest ResponseYou've already chosen the best response.0
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
 one year ago

myininayaBest ResponseYou've already chosen the best response.1
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
 one year ago

ebbfloBest ResponseYou've already chosen the best response.0
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...;)
 one year ago

UnkleRhaukusBest ResponseYou've already chosen the best response.1
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
 one year ago

ebbfloBest ResponseYou've already chosen the best response.0
@unkle agreed, but it is my preference not to do so when not necessary
 one year ago

ebbfloBest ResponseYou've already chosen the best response.0
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
 one year ago

UnkleRhaukusBest ResponseYou've already chosen the best response.1
these are some solution to differential equations ,
 one year ago

ebbfloBest ResponseYou've already chosen the best response.0
in order to make a decision as to what a student may find more clear
 one year ago

UnkleRhaukusBest ResponseYou've already chosen the best response.1
your solution must include as many arbitrary constants as the order of the differential equation
 one year ago

ebbfloBest ResponseYou've already chosen the best response.0
exactly uncle, in those solution you provided, the "y" cannot be written explicitly in terms of "x", at least not in a single expression
 one year ago

UnkleRhaukusBest ResponseYou've already chosen the best response.1
well, it can but i just looks awful , these are 'neater'
 one year ago

ebbfloBest ResponseYou've already chosen the best response.0
yeah that one can but what about (c)?
 one year ago

ebbfloBest ResponseYou've already chosen the best response.0
(e) more or less is already, not much more to do on that one...
 one year ago

UnkleRhaukusBest ResponseYou've already chosen the best response.1
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
 one year ago

UnkleRhaukusBest ResponseYou've already chosen the best response.1
implicit solutions are fine for DE's
 8 months ago
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