The constant value on the given exact differential equation

[chwala]

TL;DR

Does it matter where the constant is placed or is it placed accordingly for convenience? ...to avoid working with negative values?

Why not work with,

$y^2+(x^2+1)y-3x^3+k=0$

then,



$y^2+(x^2+1)y-3x^3=-k$

then proceed to apply the initial conditions?

My interest is on the highlighted part in red under exact_2 page.

 

[pasmith]

It's more natural to define a level surface of the conserved quantity as $F(x,y) = C$ rather than $F(x,y) = -C$; the actual sign of $C$ is of no consequence.

(The second alternative also introduces an additional minus sign, and therefore an increased risk of sign errors).

 

[chwala]

It's more natural to define a level surface of the conserved quantity as $F(x,y) = C$ rather than $F(x,y) = -C$; the actual sign of $C$ is of no consequence.

(The second alternative also introduces an additional minus sign, and therefore an increased risk of sign errors).

Thanks @pasmith . 'For convenience' as I put it...('more natural' as you put it)... or as Mathematicians like indicating 'more generally accepted...all these may apply. Cheers mate.