Linear vs Nonlinear

driven4rhythm

1. The problem statement, all variables and given/known data
I've started differential equations and I'm trying to understand the how to figure out if an equation is linear or not. The relevant equation I don't really understand either.


2. Relevant equations

3. The attempt at a solution
y' means dy/dt
1. (t^2)y'' + ty' +2y = sin(t) I said it's non linear (wrong)
2. (1+y^2)y'' + ty' +y = e^t I said linear (wrong)
3. y''''+y'''+y'' + y' + y = 1 I said linear (right) I get this because it follows along with the relevant equation from what I can tell.
4. y' + ty^2 =0 I said linear (wrong)
5. y'' + sin(t+y) = sin(t) I said non linear (right) I get this, I think, because of the sin(t+y)
6. y''' + ty' + cos^2(t)y = t^3 I said non linear (wrong)

Pengwuino

A linear equation is an equation where your derivates don't have any POWERS. It can be the 2nd,3rd, 4th, billionth DERIVATIVE, but it cant be say, [tex]y'^2[/tex] because that means the first derivative, squared, which is non-linear. Also note, [tex]y^2[/tex] is also considered a non-linear term; the functions and their derivatives can't be taken to some power >1.