We know that second differential equation is in the form y''+p(x)y'+q(x)y=g(x). To better understand seocnd differential equation, we need to know whether the equation is linear, homogeneous or non-homogeneous. Instead of providing an useless definition, here are some examples that can help you:
(y")2 + 6y' + 9y = 0 is homogeneous because all no-nconstant terms can be moved to the left side of the equation and ending up equals to 0, and you can cast the differential equation in the form (y")2 – f(x, y, y') = 0 (Notice that there's no constant terms here.
However, it is not linear because the exponent of y" is 2 instead of 1, we cannot put the differential equation into the form y''+p(x)y'+q(x)y=g(x).
d2y/dx2 + 6 dy/dx + 8y = 7 is non-homogeneous, because you can’t put all the nonconstant terms to the left side and get zero, which means you can’t cast the differential equation in the form d2y/dx2 - f (x , y , dy/dx) = 0
However it is linear because the exponent of y is 2, so we can put the differential equation into the form y''+p(x)y'+q(x)y=g(x).
Note: Notice the characters highlighted in red. Exponent of y'' and y is very important in determining whether it is linear or not!