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.

 

[blue_raver22]

Hello,

Thank you for your question. It is true that an exact differential equation involves two independent variables, while a linear differential equation involves only one variable.

An exact differential equation is one where the solution can be found by integrating both sides of the equation with respect to one of the variables. This type of equation is often used in physics and engineering to model systems with multiple variables.

On the other hand, a linear differential equation is one where the dependent variable and its derivatives appear in a linear form. These equations can be solved using various methods such as separation of variables, substitution, or using linear operators.

Both types of differential equations have important applications in various fields of science and engineering. It is important to understand the differences between them and when each type is most appropriate to use.

I hope this helps to answer your question. If you have any further inquiries, please don't hesitate to ask. Keep up the curiosity and interest in differential equations!

Best regards,