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Differentiation

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You've already dealt with vectors.

Here we just use our implicit differentiation skills.

Eventually, this capital delta

If psi is a function of x and y, and I would take not a partial derivative, I would take the full derivative of psi with respect to x, it's equal to the partial of psi with respect to x, plus the partial of psi with respect to y, times dy dx.

That's what a differential essentially is.

I expect you to see.

And we're done.

Plus the derivative of this with respect to x is easy, 2.

But this you could almost use differential notation. dy is a differential and dx is a differential.

That's what this playlist will deal with, ordinary differential equations.

So there's two big classifications.

Similarly, this is going to be the partial derivative of psi, with respect to y, but when we do the test, we take the partial of it with respect to x so we get that mixed derivative.

So plus f1 of x, that's just the first function, times the derivative of the second function.

So it's first order.

We took psi, up here, we treated the functions of x as a constant, and we just took the partial derivative with respect to y.

But anyway, so this can also be written as the partial of y, with respect to x sorry, the partial of y, and then we took the partial of that with respect to x.

So, let's see what this, this equation will do.

So what is a linear second order differential equation?

Homogeneous differential equation.

So we'll just assume that that is our psi.

They say that y of 0 is equal to negative 1.

So let's figure out what that is in terms of the derivatives of v.

So if this is an exact equation, if this is the exact equation, if we were take the partial of this, with respect to y, right?

And then you have to multiply that times the derivative of y with respect x, which is just y prime.

Or if we use calculus terminology we would view it as the derivative.

Or if you wanted to write it out, psi is this thing.

And I think this whole notion of these skills should be taught.

I just took the antiderivative of both sides.

And this is actually a separable differential equation in and of itself.

And if I wanted to take the partial of this, with respect to no, I want to take the derivative, not the partial the derivative of this, with respect to x, this is equal to the partial of psi, with respect to x, plus the partial of psi, with respect to y, times dy, dx.

Plus the derivative of this with respect to why is 3 times we're just doing the chain rule the derivative of y with respect to x.

Or the slope of the tangent line at that point of the curve.

Where psi sub x is equal to this function, is equal to 2x plus 3, and psi I shouldn't say sub x.

And this one is called homogeneous.

So let's do that.

Well that's just y prime, right?

And how do we know that?

You take the derivative of this with respect to y, you get 2.