Where is it appropriate to place absolute value bars while solving this diffeq

[vande060]

Homework Statement

just a simple problem as an example

dy/dx = y/x

dy/y = dx/x

Homework Equations

The Attempt at a Solution

dy/y = dx/x

ln|y| = ln|x| + c

then when I take exponents do I still have to include the absolute value bar

|y| = |x|c

 

[Mark44]

Homework Statement

just a simple problem as an example

dy/dx = y/x

dy/y = dx/x

Homework Equations

The Attempt at a Solution

dy/y = dx/x

ln|y| = ln|x| + c

When you exponentiate each side, you get
$$e^{ln|x|} = e^{ln|y| + c} = e^{ln|y|}\cdot e^c = Ae^{ln|y|}$$

where A = e^c

You can remove the absolute value signs by investigating four cases:
1) x > 0 and y > 0
2) x > 0 and y < 0
3) x < 0 and y > 0
4) x < 0 and y < 0

If x > 0, |x| = x
If x < 0, |x| = -x

The four cases above will probably reduce to two equations. A is always positive, so you don't have to worry about it.

then when I take exponents do I still have to include the absolute value bar

|y| = |x|c