Can Integration constant be anything? When the question is to solve a given ODE (without initial value), can I assign any value to the constant $C$, in order to solve it, or is there a specific constant value that must be found? 
In other words can two people assume different combination to the constant C to solve ODE, or one of them would be wrong?
Example: $$\frac{dy}{y}=\frac{xdx}{x^2+1}$$ $$\ln(y)=\ln\sqrt{x^2+1}+C$$
So in order to solve it I assume $C=\ln(e^c)$, but why? Can a different person assume different value? or is this the only value in the world that must be assumed for $C$ in order to solve it?
Solution:$$y=C\sqrt{x^2+1}$$

The differential equation has a solution that is unique up to multiplication by the constant $C$. In your case $C$ must be in the range of $\ln e^{c}$ with $c \in \mathbb{R}$ so $C$ can be any real number. This means you have an infinite set of solutions. If I pick something different, say my solution is $y=g(x)$ as long as there is some constant $C$ you can pick so $g(x)=C\sqrt{x^{2}+1}$ we're both okay; our functions are basically the same but mine is yours stretched or compressed vertically.
When you solve an ODE like this you're coming up with a set of solutions. In order to get a specific one you usually need an initial value for the function which will restrict your choice of $C$ to a single, distinct constant, e.g., $y(x_0)=y_0$ where $x_0$ and $y_0$ are just numbers.
