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The distinction between linear and nonlinear is pretty important
when it comes to algebraic equations,
and the same thing goes for differential equations.
So I'll explain to you in this video what a linear differential equation is,
give some examples of nonlinear differential equations,
and I'll do it by analogy with algebraic equations.
First recall what a linear equation looks like in algebra.
It looks like a combination, with constant coefficients,
of some number of unknown variables.
So here: 'a', 'b' and 'c' are some given constants
(they're known)
and 'y_1' and 'y_2' are some unknown variables,
and if I want to solve the equation
I need to find values of y_1 and y_2 which make the equation true.
A linear differential equation looks very similar:
it will be a combination of some unknown 'function', y(x)
and it's derivatives,
with some coefficients which are not constants, necessarily, but now functions
of the independent variable: in this case it's 'x'.
So 'a(x)', 'b(x)' and 'c(x)' are all some known functions
of the independent variable 'x',
and I want to find a function 'y' that makes this equation true
when I plug in y and its derivative.
Now this particular linear differential equation that I've written down here,
you may notice that the derivative only appears to the first order,
and therefore this is called a first-order linear differential equation.
There's no reason I couldn't put in more derivatives:
for example I could write down: a(x).y'' + b(x).y' + c(x).y = d(x)
(again, 'a', 'b', 'c', 'd' are some known functions of x )
and since the derivative here appears to the second order
this is called a 'second-order'
linear differential equation.
Now let me give you some examples of some nonlinear algebraic equations
and then we'll move on to nonlinear differential equations.
So how about: y_1*y_2 = 1?
(that's a nonlinear algebraic equation - it's a hyperbola)
How about: (y_1)^2 + y_2= 0? (that's a parabola)
or: [sqrt]y_1 = sin (y_2)? (that some crazy..thing)
You'll notice that none of these equations
have the form of a linear algebraic equation
(none of them look like: a.(y_1) + b.(y_2) = c. )
And by the way if I wanted to, just like I threw in an extra derivative here and
made an equation second-order,
I could throw in more variables in either the nonlinear or the linear algebraic equations
I could throw in a 'y_3' or 'y_4' if I felt like it.
Let's look at some nonlinear differential equations:
So first of all how about let's take
the 2nd derivative of y - y'' - let's square it
and set that equal to y times x?
(that's a nonlinear differential equation because I'm squaring y'')
Or how about y'*y = - x?
This is an equation that you've seen - it may look familiar,
it's nonlinear because I'm multiplying y' and y together.
Again look back at these guys and notice that
these equations cannot be put in the form
of either one of these: 'first-order' or 'second-order'.
Or how about y'' = sin (y) (again 'nonlinear'.)
Now, how about this one?
You tell me is this 'linear' or 'nonlinear'?