constant notation in differential equation

Question

Sometimes I see that $e^c$ becomes only $c$ in the solution of a differential equation. When the problem asks for the general integral there is no problem but when we have some initial condition the answer is different. I have this doubt because even Wolfram Alpha shows $e^c$ as only $c$ in the solution.

Answer 1

Consider the equation $y'(x)=y(x)$. Of course, zero function is a solution. Now let $y$ be a non-zero function. So, $$\frac{y'}{y}=1$$ and integrating we get $$\ln |y|=x+C$$ for $C\in\Bbb R$. Now, $|y|=e^C\cdot e^x$. The constant $e^C$ is positive, denote it by a new $C$. We arrive at $$|y|=Ce^x$$ for $C>0$. Because right-hand-side is always positive, then $y$ has no zeros and by continuity it has a constant sign. Then either $y=Ce^x$, or $y=-Ce^x$, where $C>0$. It allows us to denote $$y=Ce^x$$ for $C\ne 0$ ($-C$ above worked for negative constants). Having in mind that $y=0$ is a solution, we could write $$y=Ce^x$$ for $C\in\Bbb R$.

I hope it helps in your doubts.

Now put the initial condition $y(0)=-2$. Then $\ln |y|=x+C$, hence $C=\ln 2$ and $\ln |y|=x+\ln 2$. Next, $|y|=2e^x$ and, by the initial condition, we are interested with negative solutions: $y=-2e^x$.

The same could be obtained by $y=Ce^x$. Namely, $-2=C$ and $y=-2e^x$.