Hi,
Im not sure if this helps, but a linear function is a function where multiplication of the variable by a constant results in a multiplication of the function by that same constant.
Consider the function y=f(x):
y=f(x)
A*y=f(A*x)
The function is linear because multiplication of x by the constant A resulted in multiplication of y by the same constant A. OR alternately vice versa.
Example1:
y=3*x
A*y=3*x*A ???
Yes, so it is linear.
Example2:
y=x^2
We only have to disprove one case to show it is not linear, if it is not.
Set x=3, then y=9.
Set x=2*3, then y=36.
Did multiplication of x by 2 result in multiplication of y by 2 ?
The answer is no, so it's not linear.
Example3:
y=sin(x)
A*y=sin(A*x) ???
If this is not true we only have to disprove one case.
y=sin(x)
with x=pi/2 we get
y=1
With x=2*pi/2 we get
y=0
So multiplication of x by 2 resulted in multiplication of y by zero (0), so sin(x) is not linear.
More generally for any function f(x):
If:
y1=f(x)
and:
y2=f(x*A)
then:
if y2/y1=A then f(x) is linear, else it is not linear.
Someone once said, a linear differential equation is one that you can solve, a nonlinear one you can't
Often nonlinear differential equations require numerical solution.