Question

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

Answer 1

the solution given by the text is \[y=x+C(1+xy)\]

Answer 2

i believe you should cross multiply first

\[(1 + x^2)dy - (1+y^2)dx = 0\]

Answer 3

You just integrate both sides of that equation you have @ebbflo

Answer 4

You already have the variables separated

Answer 5

\[\int \frac{1}{1+z^2}\text dz=\arctan z+c\]

Answer 6

Is your trouble writing it as an algebraic expression?

Answer 7

I see all of your points and have tried those methods but do not obtain the given answer....

Answer 8

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)

Answer 9

Remember @ebbflo it does say write as an algebraic expression.

Answer 10

You get trigonometric expression right after integration

Answer 11

You want to transform that to an algebraic one using the hint I just gave you

Answer 12

Also recall tan(arctan(p))=p

Answer 13

You will use that too

Answer 14

thanks, I understand all the points you have made

Answer 15

What about the solution? Have you got the desired solution?

Answer 16

Or do you need more help?

Answer 17

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...

Answer 18

@myininaya , you can use Latex so that your math text is more clear

Answer 19

That one formula is a trig formula
The expansion for tan(a+b)
let me know if you need anything else

Answer 20

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...

Answer 21

Was your form algebraic?

Answer 22

Can you write what you have?

Answer 23

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

Answer 24

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

Answer 25

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...;)

Answer 26

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

Answer 27

@unkle agreed, but it is my preference not to do so when not necessary

Answer 28

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

Answer 29

these are some solution to differential equations ,

Answer 30

in order to make a decision as to what a student may find more clear

Answer 31

your solution must include as many arbitrary constants as the order of the differential equation

Answer 32

exactly uncle, in those solution you provided, the "y" cannot be written explicitly in terms of "x", at least not in a single expression

Answer 33

well, it can but i just looks awful , these are 'neater'

Answer 34

looking at (e)

Answer 35

yeah that one can
but what about (c)?

Answer 36

(e) more or less is already, not much more to do on that one...

Answer 37

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

Answer 38

implicit solutions are fine for DE's