Constant of integration's sign

In summary, the conversation discusses finding the solution for A in the differential equation dA/ds = -ks. The attempt at a solution involves switching the equation around and integrating, resulting in a solution of A= -(k/2)(s^2+c). However, the correct solution is given as A= -(k/2)(s^2-b^2), causing confusion about why the constant of integration must subtract from s^2. The conversation concludes by discussing the possibility of the constant of integration being complex and the need for more information to properly solve the problem.
  • #1
DocZaius
365
11

Homework Statement



Solve for A:

dA/ds = -k s

Homework Equations



See problem statement

The Attempt at a Solution



I switched the equation around:
dA= -k s ds

Integrated:

A= -(k/2) (s^2+c)

Apparently, that is wrong and I see that the answer should be:

A= -(k/2) (s^2-b^2)

My question is: Why must the constant of integration subtract from s^2? Why couldn't it add to it?
 
Physics news on Phys.org
  • #2
DocZaius said:

Homework Statement



Solve for A:

dA/ds = -k s

Homework Equations



See problem statement

The Attempt at a Solution



I switched the equation around:
dA= -k s ds

Integrated:

A= -(k/2) (s^2+c)

Apparently, that is wrong and I see that the answer should be:

A= -(k/2) (s^2-b^2)

My question is: Why must the constant of integration subtract from s^2? Why couldn't it add to it?

You wouldn't write the constant as ##-b^2## unless you had another condition given, such as an initial condition. Writing it that way implies the constant must be negative (or positive if you multiply the other minus sign through). In general you wouldn't do that. It doesn't matter whether you add or subtract the constant of integration, but it does matter if you square it, making the term positive.
 
  • #3
Couldn't the constant of integration be complex? Are you allowed to have that when s is real?
 
  • #4
If, as in your example, the functions in the DE are real and the boundary conditions are real, the solution will be real. You would typically write the solution as$$
y = -\frac{ks^2}{2} + C$$Equivalently you could write it as$$
y = -\frac{ks^2}{2} - C$$In neither case would you use ##C^2## without additional information about the DE. I thought that was what you were concerned about.
 
  • #5
It was what I was concerned about. I was trying to think of some reason why -b^2 was there, and a complex b was the only thing that could make it work, since that squared must be real and could be either positive or negative. Oh well, thank you for your help.

edit: Nevermind even the square of b could be complex if b was complex. I was thinking of the absolute value. I give up!
 
Last edited:
  • #6
DocZaius said:
It was what I was concerned about. I was trying to think of some reason why -b^2 was there, and a complex b was the only thing that could make it work, since that squared must be real and could be either positive or negative. Oh well, thank you for your help.

You're welcome. The upshot of this is that you wouldn't normally use that ##-b^2## without more info. So you were right on the money to wonder about it. There is probably more to that problem.
 

FAQ: Constant of integration's sign

1. What is the constant of integration's sign?

The constant of integration's sign is a symbol that is added to the end of an indefinite integral to represent an undetermined constant. It is usually denoted by "C" and is used to account for any possible values that the integral may have.

2. Why is the constant of integration's sign important?

The constant of integration's sign is important because it allows us to find the general solution to a differential equation. It represents all possible solutions and is necessary to fully solve the equation.

3. How do you determine the constant of integration's sign?

The constant of integration's sign is determined by evaluating the integral at a specific point and solving for the constant. This point can be chosen arbitrarily, but it is often chosen to be 0 or 1 for simplicity.

4. Can the constant of integration's sign be negative?

Yes, the constant of integration's sign can be negative. It is simply a symbol to represent an undetermined constant and its value can be positive, negative, or zero.

5. What happens if you forget to include the constant of integration's sign?

If the constant of integration's sign is omitted, the solution to the integral will be incomplete and will not account for all possible solutions. It is important to always include the constant of integration's sign when solving indefinite integrals.

Back
Top