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

 

[blue_raver22]

I understand that initial value condition questions can be challenging to grasp at first. It is important to break down the problem step by step in order to fully understand the solution. In this case, it seems that the confusion lies in the last step where the solution is given as z(x,y)= 2x-xy.

To understand where the -xy comes from, let's go back to the initial equation z(x,y)= 2x+g(xy). We have already determined that g(x)=-x, so we can substitute that into the equation to get z(x,y)= 2x-x(xy). From here, we can factor out an x to get z(x,y)= x(2-y). This is the general solution, but we can also express it as z(x,y)= 2x-xy by distributing the negative sign.

It is important to remember that in mathematics, we can express the same solution in different ways. In this case, both z(x,y)= x(2-y) and z(x,y)= 2x-xy are equivalent and valid solutions to the initial value problem. I hope this explanation helps clarify the solution for you. If you have any further questions, please don't hesitate to ask. Remember, as a scientist, it is important to continue asking questions and seeking understanding in order to further our knowledge and progress in our field.