Separable equation constant generality.

[shayaan_musta]

Hello experts!

I need your point of view about the following,

Do you think is there any loss of generality if the arbitrary constant added when a separable equation is integrated is written in the form lnC rather than just C?
Do you think this would ever be convenient thing to do?
Is there any loss of generality if the integration constant is written in the form of C²? tanC? sinC? e^x? sinhX? coshX?

Kindly clarify my concept in easiest way you can do. Use simple English, because my English is some weak.

Thanks experts.

 

[HallsofIvy]

If C is an arbitrary (but positive) constant, then ln(C) is also an arbitrary constant. As long as you careful about the signs, there will be no problem.

Perhaps you are thinking about the situation where you integrate to get ln|y|= f(x)+ C. Yes, you could replace C with ln C' (or just use C again if there is no danger of confusion) to say that $|y|= C'e^{f(x)}$. Equivalently, you could just say that taking exponentials of both sides of ln|y|= f(x)+ C to get $|y|= e^{f(x)+ C}= e^Ce^{f(x)}= C'e^{f(x)}$ where $C'= e^C$. Of course, for C any real number, $e^C> 0$. If we allow C' to be, instead, any real number, we can remove the absolute value sign.

 

[shayaan_musta]

If C is an arbitrary (but positive) constant, then ln(C) is also an arbitrary constant. As long as you careful about the signs, there will be no problem.

Perhaps you are thinking about the situation where you integrate to get ln|y|= f(x)+ C. Yes, you could replace C with ln C' (or just use C again if there is no danger of confusion) to say that $|y|= C'e^{f(x)}$. Equivalently, you could just say that taking exponentials of both sides of ln|y|= f(x)+ C to get $|y|= e^{f(x)+ C}= e^Ce^{f(x)}= C'e^{f(x)}$ where $C'= e^C$. Of course, for C any real number, $e^C> 0$. If we allow C' to be, instead, any real number, we can remove the absolute value sign.

I have better understood your thinking.

So the conclusion of discussion can be that any integration constant we use in place of C, either it is lnC, C², tanC, sinC, e^x, sinhX, coshX. Answer of the separable equation will be same as it is on just simple C?

Am I right??

 

[HallsofIvy]

A constant is a constant is a constant! It doesn't matter how you write it as long as it reduces to a number.

 

[shayaan_musta]

Ok.
I have fully understood that A constant is always a constant. It doesn't matter how you write it

But as you said

as long as it reduces to a number.

What does mean by above statement?

But in the constant list I have also shown that e^x. What will you comment about e^x? Is this can be written in the place of C or lnC, whatever I've listed?
I am asking this because I don't think so that e^x is not a constant. Because it has a variable x as a superscript, contained in the separable equation.
Am I right? Can it be written as a constant too?