Confused about expanding in a DE

[formerlyknownasprince]

Hello,
I have a separable differential equation. After integrating, I am not sure how to expand. This is what I mean:

For the second version, I left it un-expanded. It gives a different answer when I try find the constant.
Does anyone know which version to do?
Thanks in advance for the help

 

[Deleted member 4993]

Hello,
I have a separable differential equation. After integrating, I am not sure how to expand. This is what I mean:
View attachment 19345
For the second version, I left it un-expanded. It gives a different answer when I try find the constant.
Does anyone know which version to do?
Thanks in advance for the help

Those are actually the "same".

\(\displaystyle C_1 * \sqrt{7} \) = C₂

 

[formerlyknownasprince]

Those are actually the "same".

\(\displaystyle C_1 * \sqrt{7} \) = C₂

Hi there,
That is what I thought but when I graphed it on geogebra with the same initial condition, it gave two different graphs...?

 

[Deleted member 4993]

Hi there,
That is what I thought but when I graphed it on geogebra with the same initial condition, it gave two different graphs...?

Graphs of what versus what?

 

[HallsofIvy]

You have \(\displaystyle cos^{-1}(y/\sqrt{7})= \sqrt{7}(x+ c)\)
and then \(\displaystyle cos^{-1}(y/\sqrt{7})= \sqrt{7}x+ c\)

No, that's not correct- at least not for the same "c".
What is correct is that \(\displaystyle cos^{-1}(y/\sqrt{7})= \sqrt{7}x+ \sqrt{7}c\).
Of course, \(\displaystyle \sqrt{7}\) times an unknown constant is just another unknown constant but you really should use a different symbol, say "C" rather than"c":
\(\displaystyle cos^{-1}(y/\sqrt{7})= \sqrt{7}x+ C\).

Yes, "c" and "C" are different numbers.

 

[formerlyknownasprince]

You have \(\displaystyle cos^{-1}(y/\sqrt{7})= \sqrt{7}(x+ c)\)
and then \(\displaystyle cos^{-1}(y/\sqrt{7})= \sqrt{7}x+ c\)

No, that's not correct- at least not for the same "c".
What is correct is that \(\displaystyle cos^{-1}(y/\sqrt{7})= \sqrt{7}x+ \sqrt{7}c\).
Of course, \(\displaystyle \sqrt{7}\) times an unknown constant is just another unknown constant but you really should use a different symbol, say "C" rather than"c":
\(\displaystyle cos^{-1}(y/\sqrt{7})= \sqrt{7}x+ C\).

Yes, "c" and "C" are different numbers.

Ah i understand now so i would replace it with a different letter but it technically is the same, yes that makes sense thank you for your help

 

[formerlyknownasprince]

Graphs of what versus what?

Hi there, I now seem to understand your explanation before sorry bout that, thank you for your help