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

i believe you should cross multiply first
\[(1 + x^2)dy - (1+y^2)dx = 0\]

You already have the variables separated

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

Is your trouble writing it as an algebraic expression?

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

You get trigonometric expression right after integration

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

Also recall tan(arctan(p))=p

You will use that too

thanks, I understand all the points you have made

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

Or do you need more help?

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

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

Was your form algebraic?

Can you write what you have?

these are some solution to differential equations ,

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

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

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

looking at (e)

yeah that one can
but what about (c)?

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

implicit solutions are fine for DE's