ODE | Linear versus nonlinear

Examples and explanations for a course in ordinary differential equations.

In this video we explain what a linear differential equation looks like, and give some examples of nonlinear differential equations. This is done by analogy with linear and nonlinear algebraic equations. We give the general form for a first order and a second order linear differential equation.

[jossell]

3 mins compared to an hour and 25 mins in college. This guy is amazing!! Thank you!

[completelystupid]

last one is the 1st order linear differential equation

[markymark]

Correct me if I'm wrong, I think the last one is linear because it follows that form a(x)*y' + b(x)*y = c(x) where a(x) = 1, b(x) = x^2, c(x) = sin(x). None of the terms with y or a derivative of y have an exponent or sin or cos or log or anything so it is linear.

[YoLo-znbi]

I can't believe how people can make this kind of video! Awesome and I am appreciate your work

[giannacox]

Thank you for taking the time to explain this in plain english and provide examples for comparison. Very effective way to teach this. You rock!

[gwa]

The last one is a first order linear differential equation

[makasa]

Great video. Easily clears up my confusion. Thanks!
The last one is a first order linear equation

[Vonzi]

3:20 It's a first order linear non-homogeneous ordinary differential equation. So it's linear.

[jenniferchew]

Thank you so much! You seem to have a real gift for explaining concepts, and so many people, like myself, are benefiting from it. Very helpful. :)

[mohammedouallal]

Last equation is Linear, as the definition : An ordinary diferential equation y'(t) = f(t; y(t)) is called nonlinear
iff the function f is nonlinear in the second argument. ref. taken from Gabriel Nagy lecture. sin(x) is a non linear function of course but here we derive y not x.

[eismael]

First off thanks for the video, it helped to understand a lot better. I do have a question about the last one. And for y'+x^2y=sinx, why is it linear? I thought two variable mutliplying each other makes it non linear?

[luvrajkhadkabk]

3:15 why did he say double prime when it is single prime? Also are you sure its non-linear?

[claremcdermott]

Just to clarify.. is a linear ODE where the dependent variable appears in a linear fashion? 

[mohammedouallal]

Short, clear, nice! This guy apply what we call 'Consistency'!

[mokarrommolla]

"That's some crazy thing" lol

[hybmnzz]

These are basically linear combinations of the derivatives of y (where the 0th derivative is just y itself) except the scalars are really scalar functions. This is why you see linear algebra applied to ODEs sometimes.

[blckmmbfrlif]

I believe all the y terms involved in each term of a differential equation is only supposed to contain a just 1 y. Like
f(x)y'' + g(x)y' + h(x)y = j(x)
Notice the only non derivative y allowed in linear is y^1.
And Neither of the y terms are involved in any form of product of y.

[Latripleline]

Out of topic here but what makes y1y2 = 1 a hyperbola ? Theres neither y^2 nor x^2 in that equation how did you find out?

[abhishekgujjar]

is x^3 (Y2)^2+x (Y1)^4=0 a linear equation or a nonlinear

[TheDua]

it is linear Diff. Equation, is it ??