Why are these differential equations linear or nonlinear?

[jaejoon89]

In class, my teacher gave the following equations as examples of linear and nonlinear ODE. In the first equation, there are x's in front of some of the y's yet it is linear. In the second equation, there is an x in front of y^2 yet it is nonlinear - why? Also, why is the final equation nonlinear?

linear

x^2 y''' + (x-1)*y'' + sin(x)*y' + 5*y = tan(x)


nonlinear

y' + x*y^2 = 0
y'' + sin(x+y) = sin(x)

 

[LCKurtz]

They are linear because they are first degree in the dependent variable and its derivatives. No y², e^y, etc., and the same for y', y'' etc.