Separable equation constant generality.

In summary, the conversation discusses whether there is 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. The experts clarify that as long as the signs are taken into account, there will be no problem regardless of how the constant is written. They also explain that any integration constant can be used in place of C and the answer will still be the same. The experts also mention that ex can also be used as a constant in place of C or lnC, despite having a variable as a superscript in the separable equation.
  • #1
shayaan_musta
209
2
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? ex? sinhX? coshX?

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

Thanks experts.
 
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  • #2
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 [itex]|y|= C'e^{f(x)}[/itex]. Equivalently, you could just say that taking exponentials of both sides of ln|y|= f(x)+ C to get [itex]|y|= e^{f(x)+ C}= e^Ce^{f(x)}= C'e^{f(x)}[/itex] where [itex]C'= e^C[/itex]. Of course, for C any real number, [itex]e^C> 0[/itex]. If we allow C' to be, instead, any real number, we can remove the absolute value sign.
 
  • #3
HallsofIvy said:
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 [itex]|y|= C'e^{f(x)}[/itex]. Equivalently, you could just say that taking exponentials of both sides of ln|y|= f(x)+ C to get [itex]|y|= e^{f(x)+ C}= e^Ce^{f(x)}= C'e^{f(x)}[/itex] where [itex]C'= e^C[/itex]. Of course, for C any real number, [itex]e^C> 0[/itex]. 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, ex, sinhX, coshX. Answer of the separable equation will be same as it is on just simple C?

Am I right??
 
  • #4
A constant is a constant is a constant! It doesn't matter how you write it as long as it reduces to a number.
 
  • #5
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 ex. What will you comment about ex? 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 ex 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?
 

1. What is a separable equation?

A separable equation is a type of differential equation in which the variables can be separated into two different functions, allowing for an easier solution.

2. What is a constant of integration?

A constant of integration is a term that is added to the solution of a separable equation to account for any indefinite integrals that may arise during the solving process.

3. How do you solve a separable equation?

To solve a separable equation, you must first separate the variables, then integrate both sides, and finally add the constant of integration to the solution.

4. What is the general solution to a separable equation?

The general solution to a separable equation is the most general form of the solution, including the constant of integration. It is typically denoted by C and can be used to find specific solutions for different initial conditions.

5. What is the advantage of using separable equations?

The advantage of using separable equations is that they can often be solved using basic integration techniques, making them easier to solve than other types of differential equations.

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