# Linear First-Order Differential Equation

1. Dec 7, 2015

### DiamondV

Hi,
so I am just starting to learn differential equations and we were told that you have to put the equation into
this form.

In one of the examples in my lecture notes, which says
"Solve (x^2-9)dy/dx+xy=0, x>3" He then says we need to rearrange this into the form given above, so he does this by dividing across by x^2-9 and gets this.

I understand that if you divide 0 by x^2-9 you get 0 but that still isnt a function of x

Now if you compare the form we put the equation into and the required form you will notice that

My issue is with the f(x). How is 0 a function of x? Arent we meant to have a function of x on the right as part of the required format?

2. Dec 7, 2015

### PeroK

If you are unhappy with $0$ just replace it with $x - x$, which perhaps you'll accept as a function of $x$.

3. Dec 7, 2015

### Staff: Mentor

$f(x) = λx$ is a function of $x$ for every $λ$, a straight through the origin. With $λ=0$ it just becomes the x-axis, but is still a straight through the origin and a function of $x$. Also $f(x) =$any constant is a function of $x$.

4. Dec 10, 2015

### HallsofIvy

Staff Emeritus
A constant function, f(x)= any number is also a function! The basic definition of "function" is "a set of ordered pairs such that no two pairs have the same first value". The function f(x)= 0 for all x is the set of ordered pairs {(x, 0)} where x can be any number.