Differential Equations Constants

Question

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:

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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.

Answer 1

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.

Answer 2

i think the problem lies in this line $$-\frac{1}{5}\ln(y)c_2=t+c_1$$ and then we have $$-\frac{1}{5}\ln(y)=t+c_1-c_2$$ then you can define $$C=c_1-c_2$$