Original subtitles

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Hey everyone welcome back.

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And this video we're going to talk about the difference between what's called linear differential equations

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vs. nonlinear linear versus non-linear.

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This topic is a little bit harder for some people.

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So we'll do a lot of examples.

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So A D E so a differential equation is linear in Y.

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You can also say linear in X or any other variable we're going to focus on linear in y if we can write

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it.

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So if we can write it in the form.

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So first I'm going to show you you like the mathematical definition and then I'm going to explain it

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in words because that's how I think of it.

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I actually don't think of it this way.

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I mean I guess I do.

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Let me just go ahead and write it.

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So basically you have a coefficient.

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So I'll call that a sub end of X so a pure function of x times say the derivative of x plus and then

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you would have a sub N minus one.

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But I'll skip that one all the way down to say the first derivative.

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So a sub 1 of X D Y D X again a pure function of x and then plus a sub zero of X times the zero with

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derivative.

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So the zero is derivative of a function is the original function and that should be equal to G of X.

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So basically in front of y and all of its derivatives.

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So here's why.

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And all of its derivatives you only have pure functions of x.

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That's the first condition.

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So again in front of Y in all of its derivatives you only have pure functions of x and the second condition

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is that y in all of its derivatives are all to the first power so maybe you should write that down so

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1 so in front

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of y and its derivatives

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you have only pure functions of acts so pure functions of x.

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Also here you have the only X here.

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Right can't have a Y here but in front of y and all of its derivatives you have pure functions of x

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and then also y and its derivatives

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are altered the first power.

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So are all to the first power OK let's do some examples right away and we're gonna do a bunch of these

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and so hopefully it makes sense.

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It's a question.

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It says linear or nonlinear so linear in Y.

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We're only concerned if it's linear or non-linear and Y or is it nonlinear nonlinear.

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OK.

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A How about let's say I have to make these up.

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How about y double prime plus.

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Each of the X times Y prime plus Y equals sine X.

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So this is a differential equation.

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If you watch the first video you'll know that the order is two because the order is the order of the

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highest derivative and there's a second derivative in this problem.

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We want to figure out if it's linear or nonlinear.

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So here we have a one.

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Here we have an e of the X.

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Here we have a one so in front of y and all of its derivatives.

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We have pure functions of x that that's the first condition check and Y.

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And all of its derivatives are all to the first power.

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So that's the second condition check.

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And then here we have a pure function of x so check.

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So in this case the answer would be linear linear.

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Let's do another one.

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B How about this one y triple prime plus three y y double prime plus Y equals each of the x.

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In this case the order is three.

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Just easy to talk about because the order is the order of the highest derivative.

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And this one would be nonlinear because of this piece right here.

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Right.

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This is a function of y and it's being multiplied by the second derivative.

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You cannot have that right.

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You have to have only pure functions of x in front of y and its derivatives.

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So in this case the answer would be nonlinear nonlinear.

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Let's do another one.

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C How about something like this.

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X squared away the X plus Y equals seven OK.

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So in this case the order is one because the order is the order of the highest derivative.

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So we have the first derivative and this one won't be linear because this is a pure function of X. There

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is a one here.

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It's a pure function of x and y and all of its derivatives are to the first power.

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So that would be linear.

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Keep going D How about something like this square root.

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Do you y the x plus three y equals sine X in this case the order is 1 because this is the first derivative

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and this one is nonlinear.

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And the reason is you can think of this D Whitey X this piece here rather as do y the X to the one half

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power right.

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Because you can write the square root as a 1 1/2.

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So that's not the first power.

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So this is non-linear right.

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Y and all of its derivatives all have to be to these first powers.

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Otherwise it's nonlinear e y double prime plus sine y times Y equals E to the.

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This wouldn't be order to right because the order is the order of the highest derivative and it would

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be nonlinear because you have this function of y here in front of the Y.

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He can't have that right.

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No good at all.

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So this is non-linear you can only have pure functions of x in front of y and all of its derivatives.

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One more f see if I have any ideas I can't really think of anything that we haven't really done too

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much.

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Well how about this Y double prime plus cosine of X plus Y equals each of the x.

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So kind of a weird problem here but this is a differential equation and it's order 2 because the order

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is the order of the highest derivative and that's a second derivative it's been non-linear.

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Right non-linear you cannot have cosine of X plus Y doesn't make any sense in this context right.

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Y and all of its derivatives have to be to the first power it's inside a cosine function so it's not

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good.

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That would be non.

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So hopefully this medium video made sense and just keep in mind it's linear.

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If Y and all of its derivatives appear to the first power and in front of y and all of its derivatives

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you have to have pure functions of x on the right hand side you always have a pure function of X..

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I hope that helps.