Question

How do you know where to put the constant when finding general solutions for differential equations?

Answer 1

Let's suppose we want to solve $y'=y$ with $y(0)=2$. Separating variables and integrating, we get

$\displaystyle\int\dfrac{\mathrm dy}y=\int\mathrm dx\implies\ln|y|=x+C\implies y=e^{x+C}$

Leaving the solution in this form, the initial condition gives

$2=e^{0+C}=e^C\implies C=\ln2$

This means the solution is $y=e^{x+\ln2}$.

Now if we were to write $y=e^{x+C}=e^xe^C=Ce^x$, then we would have found

$2=Ce^0\implies C=2$

so that the solution would have been $y=2e^x$.

But these two solutions are the same, since $y=e^{x+\ln2}=e^xe^{\ln2}=2e^x$. So we get the same solution regardless of where we place $C$, despite getting different values for $C$.