Differential Equations: Understanding Linnearity

[hussness]

I have been wondering ever since the first day of my differential equations class what linnearity means intuitively for a differential equation. I remember first being taught that linnearity meant an equation could be put in the form:

y = mx + b

Is the DE definition of linnearity at all related to the agebraic definition or am I getting mixed up by equivocation?

I would also like to get an idea of what various elementary functions look like so that when I read a problem it's not just letters on a page. Any suggestions for how I might best go about doing this?

 

[neutrino]

I think it refers to the fact that the solutions of "simple" ODEs obey the principle of superposition, i.e. a linear combination of solutions is also a solution.

 

[d_leet]

Well a linear equation in two variables as you ahve seen in algebra is one of the form

Ax + By = C

And a first order linear differential equation is of the form

p(x)y + q(x)y' = r(x) where p ,q and r are functions of x, then this equation can be said to be linear in the variables y and y', I think I saw it presented this way in the first lecture video for the differential equations course on MIT's opencourseware site.

 

[d_leet]

I would also like to get an idea of what various elementary functions look like so that when I read a problem it's not just letters on a page. Any suggestions for how I might best go about doing this?

Are you asking what the graphs of the elementary functions look like? Or are you asking what the elementary functions are?

 

[hussness]

I'm talking about the graphs.

Let's take a simple example of a LDE:
y' = cos(x)
A particular solution would be y = sin(x)
Neither sin(x) nor cos(x) are linear functions according to my understanding. How then can we say that y' = cos(x) is a linear DE?

 

[d_leet]

I'm talking about the graphs.

Let's take a simple example of a LDE:
y' = cos(x)
A particular solution would be y = sin(x)
Neither sin(x) nor cos(x) are linear functions according to my understanding. How then can we say that y' = cos(x) is a linear DE?

Because linearity of a differential equation has nothing whatsoever to do with whether or not the solution to that differential equation is linear, I'm not exactly sure how to explain this other than how I did in my first post so hopefully someone will come and answer that better than I did.

 

[HallsofIvy]

Definition of "linear differential equation": there are only linear functions of the dependent variable and its derivatives. In y'= cos(x) the dependent variable is y not x. The only function of a derivative of y is y' itself which is linear (the function f(x)= x is linear).

The basic idea of linear problems in general is that we can "decompose" them- break the problem into parts, solve each part, then reassemble the solutions into a solution to the entire problem.

 

[hussness]

Thank you. That clears things up a bit.