solution to an ODE separable variable

Question

I have this ODE: $du/u = dt$ which I think the solution is $\ln|u| = t + c \leftrightarrow |u| = e^{t+c} = ce^t$. Is this correct? Is $u$ really a function then, since each value of $t$ would be mapped onto two values of $u$?

Answer 1

Here are the steps $$ \frac{1}{u}du=dt $$ $$ \int \frac{1}{u}du=\int dt $$ $$ \ln |u|+C_1=t +C_2 $$ $$ \ln |u|=t +C_2 -C_1=t+C $$ $$ e^{\ln |u|}=e^{t+C} $$ $$ u=e^{t+C} $$ Note that the domain of $\frac{1}{u}$ is $\{u\in\mathbb{R}:u\ne 0\}$, which includes negative numbers. Also the domain of $\ln u$ is $\{u\in\mathbb{R}:u\gt 0\}$, which does not include negative numbers. This is why when we integrate $\frac{1}{u}$, we get $\ln|u|+C$. Once we get rid of the $\ln|u|$, we can also get rid of the absolute value bars because $u$ in this case, is allowed to be negative.