Q:

Differential Equations Constants

Help! I'm going nuts with all the constants! consider this equation: $y'+5y=0$. When you getting to the integral part: $\int y^{-1}\,dy=\int-5\,dx$. After the integration $\ln(y)+c_1=-5x+c_2$. Now we have 2 $c$'s. When I use a calculator I don't know where the $c_2$ disappears:

$%[1]https://screenshotscdn.firefoxusercontent.com/images/4b88830c-771e-4b55-9c27-4a111c13b941.png.$

They write combine the constants. This is not changing the result? And how do I know it's $y=-5x+c$ and not $y=-5x-c$? Thanks.

A:

There's something weird with the provided solution. Simply :
$$y'+5y = 0 \Leftrightarrow \frac{y'}{y}=-5 \Rightarrow \int \frac{y'}{y}dx = \int -5dx \Leftrightarrow \ln |y(x)| = -5x+c $$
$$\Rightarrow$$
$$|y(x)| = e^{-5x+c} =e^{-5x}e^c$$
Now if you simply let $c := e^c$, you yield the expression  :
$$|y(x)| = ce^{-5x}$$
due to the fact that the constant $c \in \mathbb R$ is arbitrary so you can "replace" it to whatever you like.

