Let's suppose we want to solve [tex]y'=y[/tex] with [tex]y(0)=2[/tex]. Separating variables and integrating, we get
[tex]\displaystyle\int\dfrac{\mathrm dy}y=\int\mathrm dx\implies\ln|y|=x+C\implies y=e^{x+C}[/tex]
Leaving the solution in this form, the initial condition gives
[tex]2=e^{0+C}=e^C\implies C=\ln2[/tex]
This means the solution is [tex]y=e^{x+\ln2}[/tex].
Now if we were to write [tex]y=e^{x+C}=e^xe^C=Ce^x[/tex], then we would have found
[tex]2=Ce^0\implies C=2[/tex]
so that the solution would have been [tex]y=2e^x[/tex].
But these two solutions are the same, since [tex]y=e^{x+\ln2}=e^xe^{\ln2}=2e^x[/tex]. So we get the same solution regardless of where we place [tex]C[/tex], despite getting different values for [tex]C[/tex].
