Question concerning constants of integration

[Duderonimous]

Homework Statement

How can one simply let C=lnk? Thus changing

y=\pm\sqrt{ln(t^{2}+1)+C}

to

y=\pm\sqrt{ln[k(t^{2}+1)]}

Homework Equations

None

The Attempt at a Solution

I know they are both arbitrary constants, are there restrictions on the allowed values of the constants? Actually I checked the answer in the book and it said k is allowed to be any positive real number. I understand because it is under the radical. Insight into this would be helpful.

 

[pasmith]

I know they are both arbitrary constants, are there restrictions on the allowed values of the constants? Actually I checked the answer in the book and it said k is allowed to be any positive real number. I understand because it is under the radical. Insight into this would be helpful.

For every real C, there exists a unique k > 0 such that C = \ln k. Thus one can always replace an arbitrary constant C with \ln k where k > 0 is arbitrary.