Need explanation of this differential equation

[aruwin]

I need explanations at the last part of this math solution.

Question:
Solve the differential equation:
y' = (1 + 2/x)y

Answer:
ln|y| = x+ln(x^2)+c

|y| = e^c.x^2.e^x

y = Cx^2.e^x (C = +/-e^c is any constant that is not equals to 0)

What I don't understand is this part where : |y| = e^c.x^2.e^x
Why do we have to multiply all the terms when we take the "In" out?

 

[Simon Bridge]

Because what you do to one side has to be done to the other and $a^{(b+c)}=a^ba^c$ ... step by step:

starting from:
$\ln|y|=x+\ln(x^2)+c$ ... take the exponential of both sides:

$y=e^{x+\ln(x^2)+c}=e^x e^{\ln(x^2)}e^c$

 

[HallsofIvy]

The symbol is not "In" it is "ln" for logarithm.