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.