Question

Diff EQ.
Could someone explain the where the values for N(y) and M(x) are coming from in this first order separable equation? Look at attachment, the original equation is highlighted at top, and my question highlighted at bottom.

Answer 1

that's a freak show, IMHO :-))

you have \(\dot y^2 = 1- y^2\)

so \(\dot y = \sqrt{1-y^2}\), .... strictly speaking..... \(\pm\) ... but run with it for a while


so \(\dfrac{dy}{\sqrt{1-y^2}} = dt\)

and you go from there with trig and stuff

Answer 2

Like what HUlaHOop said, just make a comparison between the answer and the standard form, you can see where N(y) , M(x) come from
\(\color{red}{N(y)} dy = \color {blue}{M(x)} dx\\\color{red}{\dfrac{1}{\sqrt{1-y^2}}} dy = \color {blue}{1} dx\)

Answer 3

but why is it -1/(sqrt(1-y^2)). The negative is throwing me off

Answer 4

Would you just assume it has to be negative, because you want your final answer y=-sin(t), and just throw out the possibility of getting the positive sin(x) as your answer(which by the way I've checker and positive sin(x) couldn't be solution.
Anyway thank you for the help : )

Answer 5

you're battling in a bathtub here, dude

try the solution y = + sin (whatever) as opposed to y = - sin (whatever)

experiment🙂