# First ODE of an absolute value

1. May 3, 2012

### sydneyw

so I understand the basic premise of differentiating a first ODE, or I thought I did. I have the equation y'-y=abs(x-1). I have no idea of how to go about this. Can someone walk me through how to do this? I'm attempting to study for a test and this is one of the practice questions he gave us so I feel as though I'm in some serious trouble if I don't learn how to do this! Thank you much.

2. May 3, 2012

### LCKurtz

Can you solve $y'-y=x-1$? Can you solve $y'-y=1-x$? These are the two cases you have depending on whether $x>1$ or $x<1$. Solve them separately.

3. May 3, 2012

### HallsofIvy

Staff Emeritus
First, you don't want to "differentiate" the ODE, you want to integrate it.

And the simplest way to handle the absolute value is to use the definition. If $x\ge 1$, x- 1 is non-negative so |x- 1|= x- 1 and your differential equation becomes dy/dx- y= x- 1.
If x< 1, x- 1 is negative so|x- 1|= -(x- 1)= 1- x and your differential equation becomes dy/dx= 1- x.

Integrate those to get two general solutions, one valid for x> 1, the other valid for x< 1.