Let's do some more nonhomogeneous equations.

That this is a solution to our inhomogeneous equation, and that any solution to our inhomogeneous equation takes this form right here.

So the inhomogeneous equation looks like this.

You essentially solve the homogeneous equation.

And if we look at our original problem, the general solution out of this nonhomogeneous equation would be this which is the general solution to the homogeneous equation, which we've done many videos on plus now our particular solution that we solved using the method of undetermined coefficient.

But what about the case when you're solving the inhomogeneous equation.

Now, we just said that any solution to an inhomogeneous

You get the general solution of the homogeneous.

Maybe we'll call this the homogeneous solution. y homogeneous.

And that's all and good, but in order to get the general solution of this nonhomogeneous equation, I have to take the solution of the nonhomogeneous equation, if this were equal to 0, and then add that to a particular solution that satisfies this equation.

Because I think you know how to solve the essentially, the homogeneous version.

Homogeneous differential equation.

So anyway, let's take these and substitute them back into this nonhomogeneous equation.

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So we solved the homogeneous equation.

What does a homogeneous differential equation mean?

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And that's really what you're doing it the method of undetermined coefficients.

Let's do one more homogeneous differential equation, or first order homogeneous differential equation, to differentiate it from the homogeneous linear differential equations we'll do later.

Now why am I so concerned with this, and I've been kind of fixated on this inhomogeneous equation for some time.

Far from it. The same disproportionate patterns of school assignment exist throughout much of Central and Eastern Europe.

I will now introduce you to the idea of a homogeneous differential equation.

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Schwannomas are homogeneous tumors, consisting only of Schwann cells.

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So the traditional ways of separating materials just simply don't work for plastics.

And we said in the last video that you can kind of given this you can kind of think of the solution set to the inhomogeneous equation is equivalent to some particular solution, let's call that x particular, some particular solution, plus some member of your null space.

And now in the last example, the nonhomogeneous part was 3e to the 2x.

However, it is not capable of providing food supplements.

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I don't believe that chemicals are unnatural.

Such differences can be more significant than the differences between groups, especially in the case of heterogeneous populations and value systems.

. As sanctions are being lifted, South Africa has come under increasing international pressure to discontinue its unwieldy and sometimes inconsistent trade policies.

Those are called homogeneous linear differential equations, but they mean something actually quite different.

But anyway, for this purpose, I'm going to show you homogeneous differential equations.

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But that is more difficult when neighboring countries are exploiting the country's heterogeneous makeup to pursue their own hegemonic agendas.

But even this monolithic system of control is porous.

That is going to be equal to b and that this is going to be equal to the 0 vector because this is a solution to our homogeneous equation.

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We said that h is a solution for the homogeneous equation, or that this expression is equal to 0.

At the nanometer scale (about 50,000 times smaller than the width of a human hair), matter behaves in unusual ways weak materials become strong, inert materials become active, and benign materials become harmful.

I'm also interested in working with unusual materials and common materials in unusual ways.