Original subtitles

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Everyone in this video were briefly going to talk about what's called initial value problems so initial

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value problems.

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I just wanna explain the concept behind them and what they mean.

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So high.

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V.P. is an easy way to abbreviate initial value problem.

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Due to examples.

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So Example 1 say we have a differential equation so d y the X and it's equal to some stuff over here

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on the right hand side with Xs and Ys so to indicate that I'm gonna to write F of X comma Y.

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So all this means is that we have some stuff with X's and Y's that's all that notation means.

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Okay.

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So like X plus Y etc. X squared plus sine Y etc. I just stop with X and Y then we have a condition Y

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of x sub zero equals Y some zero.

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This is called an initial condition.

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So initial condition and typically I like to revisit it as I see.

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So that's our I see s our initial condition.

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So what does this mean graphically.

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Well if you solve a differential equation you know most of the time you get like you know X squared

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plus C or something like that you get a constant.

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In this case you would get one arbitrary constant so you might get like you know infinitely many solutions

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so something like this.

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And then something like this something like this one for each choice of C right you have infinitely

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many solutions.

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And so what's happening here is that this initial condition well what does this mean.

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This means that the x coordinate is X not on the y coordinate is why not.

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So this means that your solution passes this point.

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So what it's doing is it's picking a particulates solution from the infinitely many solutions.

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So like maybe it's picking this one.

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So this red graph here this this would be the solution

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to the AVP

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so whenever you have one arbitrary constant and your solution you haven't wanted whenever you have a

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one parameter family to find a particular solution you would need one initial condition.

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In this case the initial condition tells us graphically that our solution passes the point.

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So we have infinitely many solutions and then we're picking a particular one.

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Right.

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Maybe that's what the name particular comes from right.

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Because once you use this you're gonna find your C so you'll get something like Y equals.

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Making this up but X squared plus four let's say.

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So that might be your answer.

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So this is called a particular solution because it has no C's.

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So that's that's an initial value problem.

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Here's another example.

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Say we had DIY D X equal to now actually mean do a second derivative here.

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There we go.

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Equal to some stuff with X Y and the first derivative.

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So now we have an order to differential equation so X Y and then the first derivative.

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So this is just some stuff with X Y and the first derivative.

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So now we need to initial conditions maybe y of x sub zero equals Y sub zero and then Y prime of x sub

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zero equals Y sub one equals Y sub 1.

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So now again say we have you know infinitely many solutions you know something like this

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right.

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And then what this does is it obviously it picks the one that passes through through X not why not.

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So it picks that one.

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So maybe that would be this one here so it would pick that one but not only that it guarantees that

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the slope at this point is is this right this slope slope is why 1 the slope of this tangent line is

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why once the slope of the function or the tangent line at that point is equal to that.

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So this requires two things.

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Right.

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It passes through this point and it has this particular slope at that point.

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So in this case if like if you were to solve this d e maybe it would be something like I'm just gonna

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make this up.

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C 1 each of the X plus C 2 x squared was totally making this up.

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So you'd have to seize and see what you would have to use both of these conditions to find both CS and

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when you do that you would get the particular solution.

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So in this case the particular solution would be would be that right there.

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So I hope that made sense.

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I'm going to make another video showing you a little bit more how to how to actually find the seas and

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stuff like that.

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That's it.