The solutions to this algebraic equation are going to be numbers, or a set of numbers. We can solve this, it's going to be x plus two times x plus one is equal to zero. So x could be equal to negative two or x could be equal to negative one. The solutions here are numbers, or a set of values that satisfy the equation.
Here it's a relationship between a function and its derivatives.
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And so the solutions, or the solution, is going to be a function or a set of functions. Now let's make that a little more tangible. What would a solution to something like any of these three, which really represent the same thing, what would a solution actually look like? Actually let me move this over a little bit.
Move this over a little bit. So we can take a look at what some of these solutions could look like. Let me erase this a little. This little stuff that I have right over here. So I'm just gonna give you examples of solutions here.
We'll verify that these indeed are solutions for I guess this is really just one differential equation represented in different ways. But you'll hopefully appreciate what a solution to a differential equation looks like. And that there is often more than one solution. There's a whole class of functions that could be a solution.
So one solution to this differential equation, and I'll just write it as our first one. So one solution, I'll call it y one. And I could even write it as y one of x to make it explicit that it is a function of x. One solutions is y one of x is equal to e to the negative three x.
Power Geometry in Algebraic and Differential Equations, Volume 57
And I encourage you to pause this video right now and find the first derivative of y one, and the second derivative of y one, and verify that it does indeed satisfy this differential equation. So I'm assuming you've had a go at it. So let's work through this together. So that's y one. So the first derivative of y one, so we just have to do the chain rule here, the derivative of negative three x with respect to x is just negative three.
And the derivative of e to the negative three x with respect to negative three x is just e to the negative three x. And if we take the second derivative of y one, this is equal to the same exact idea, the derivative of this is three times negative three is going to be nine e to the negative three x. And now we could just substitute these values into the differential equation, or these expressions into the differential equation to verify that this is indeed going to be true for this function. So let's verify that. So we first have the second derivative of y.
Department of Mathematics: Fields of Research
So that's that term right over there. So we have nine e to the negative three x plus two times the first derivative. So that's going to be two times this right over here. So it's going to be minus six, I'll just write plus negative six e to the negative three x. Notice I just took this two times the first derivative. Two times the first derivative is going to be equal to, or needs to be equal to, if this indeed does satisfy, if y one does indeed satisfy the differential equation, this needs to be equal to three times y.
Elements of Commutative Algebra and Algebraic Geometry
Well three times y is three times e to the negative three x. Three e to the negative three x. Let's see if that indeed is true. So these two terms right over here, nine e to the negative three x, essentially minus six e to the negative three x, that's gonna be three e to the negative three x. It seems that you're in Germany. We have a dedicated site for Germany.
This book is a short primer in engineering mathematics with a view on applications in nonlinear control theory. In particular, it introduces some elementary concepts of commutative algebra and algebraic geometry which offer a set of tools quite different from the traditional approaches to the subject matter. This text begins with the study of elementary set and map theory.
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Chapters 2 and 3 on group theory and rings, respectively, are included because of their important relation to linear algebra, the group of invertible linear maps or matrices and the ring of linear maps of a vector space. Homomorphisms and Ideals are dealt with as well at this stage. Chapter 4 is devoted to the theory of matrices and systems of linear equations.
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Chapter 5 gives some information on permutations, determinants and the inverse of a matrix. Chapter 6 tackles vector spaces over a field, Chapter 7 treats linear maps resp. Chapter 8 considers the diagonalization of a matrix and their canonical forms.
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