I What is the notation for referencing an ODE with different parameter values?

[member 428835]

Hi PF!

Suppost I had some differential equation, say $$y'(x) + axy(x) +7a =0$$ where $y=f(x)$ and $a$ is some parameter. How do you reference this differential equation with different $a$ values are plugged in? Would I say $$ F(x,y;a) \equiv y'(x) + axy(x) +7a =0$$ and then when referencing an $a$ value of $12$ simply say $F(x,y;12)$?

Thanks

 

[Simon Bridge]

The LHS is a specific instance of $F(x,y(x),y'(x))$ if you want to characterize it by a key parameter, then you write $F_a(x,y,y')$ for discrete parameters, and $F(x,y,y',a)=0$ for continuous parameters. There is no functional difference between a parameter that is allowed to vary and any other kind of variable.

I don't think there is a universal notation though - define your own. i.e. if F is defined to be y'+axy+7a then F(a) is enough to specify the particular F you mean.

 

[member 428835]

Thanks a ton! This is all I wanted!

 

[Simon Bridge]

I forgot one: in statistics there is a standard notation ... $f(x|a,b)$ signifies a general function of x with parameters a and b, which have to be specified.
The pipe character "|" reads "given" and you also see it in conditional probability statements.