Confusion about linear equations

[bentley4]

Hi everyone!

1. Is a linear equation the same as a polynomial of first(or 0th) degree?
2. The book 'Mathematics for physicists and engineers' by springer(publisher) states that an example of a linear (DE) equation is 5.dy(x)/dx = x.y(x). Yet I read somewhere else that f(x,y)=a.x.y+b is not linear.(polynomial of 2nd degree).
Is one them wrong? Or are both right because in the first example y is a depent variable while in the second example y is an independent variable?

 

[Vorde]

I think you have a language mix up here. The first equation is a linear differential equation, which means something different than linear equation in the basic algebraic sense.

 

[Fredrik]

If you denote the operator that takes a differentiable function f to its derivative f' by D, then your differential equation can be written as Df(x)=xf(x), or equivalently, as (D-Q)f=0, where I have defined a new operator Q by Qf(x)=xf(x). The equation (D-Q)f=0 is said to be linear because the operator D-Q is linear.

The function f defined by f(x,y)=axy+b is clearly not linear.


(Recall that a function T from a vector space to a vector space is said to be linear if T(ax+by)=aTx+bTy for all scalars a,b and all vectors x,y).