12-02-2014, 11:30 PM
In a differential equation, when the variables and their derivatives are only multiplied by constants, then the equation is linear. The variables and their derivatives must always appear as a simple first power. Here are some examples.
x'' + x = 0 is linear
x'' + 2x' + x = 0 is linear
x' + 1/x = 0 is non-linear because 1/x is not a first power
x' + x2 = 0 is non-linear because x2 is not a first power
x'' + sin(x) = 0 is non-linear because sin(x) is not a first power
x x' = 1 is non-linear because x' is not multiplied by a constant.
x'' + x = 0 is linear
x'' + 2x' + x = 0 is linear
x' + 1/x = 0 is non-linear because 1/x is not a first power
x' + x2 = 0 is non-linear because x2 is not a first power
x'' + sin(x) = 0 is non-linear because sin(x) is not a first power
x x' = 1 is non-linear because x' is not multiplied by a constant.