Differential Equations - 1 vs 2 Variables?

[amaresh92]

greetings,
is it true that a exact differential equation is for two independent variable whereas linear differential equation is for one variable?

advanced thanks,

 

[HallsofIvy]

greetings,
is it true that a exact differential equation is for two independent variable whereas linear differential equation is for one variable?

advanced thanks,

No, it's not. The fact is that it is so easy to switch a first order d.e. from "dy/dx" to "dx/dy" that it is better, for most first order differential equations, not to think of one variable as being the "independent" variable and the other variable as "dependent". That may be what you mean when you talk about "two independent variables". You can think of either one as "depending" on the other.

The exception to that is something like $dy= [(2x^2y+ sin(x))]dx$ where if we think of y as a function of x we have a linear equation whereas, while we can think of x as a function of y, the equation is no longer linear and so not as easy to solve.

 

[Delta2]

Hm maybe you have in mind a DE that can be written in a form df=0, where f is a function of x,y that is f(x,y) so x and y can be considered independent variables of the function f (so that you can take partial derivatives wrt x and y and such).

But the solution to df=0 is f(x,y)=c so in a sense x and y are not independent variables but they are connected by the equation f(x,y)=c.

 

[HallsofIvy]

What is really true is that it is better to think of x and y as depending on one another and not think of anthing as an "independent" or "dependent" variable. I see no difference between the two examples you give.