Trouble with Initial Value Condition Questions

[porroadventum]

I have been looking at an example of a initial value condition problem in my notes and don't really understand where the solution came from. Here is the question:

Let z(x,y)= 2x+ g(xy) and add the initial value conditon, z= x on the line y=1. Find the general solution of the initial value problem.


1. Replace z(x,y)=2x+g(xy) wih the condition to get x= 2x+g(x) for all x, so that g(x)= -x

I understand everything so far but then the next step says "hence z(x,y)= 2x-xy is the general solution." Where does the -xy come from?

Any help or advice would be much appreciated! Thank you

 

[Simon Bridge]

Can you find another relation that would fit the conditions?

 

[HallsofIvy]

It's just a matter of the "functional notation" you have been using for years:

If g(x)= -x then g(u)= -u, g(a)= -a, g(v)= -v, etc.

In eactly the same way, g(xy)= -xy.

 

[porroadventum]

OK I understand now, thank you