Search results

  1. 5

    Second order non-linear differential equation involving log

    awesome. Also, I also think you are correct about the missing C, thank you--I shall update that post. Thank you for your help!
  2. 5

    Second order non-linear differential equation involving log

    true. OK, then \displaystyle{\frac{dv}{dx} = \frac{v \log(v) - v}{x} \Longrightarrow \int{\frac{1}{v} \cdot \frac{1}{\log(v) - 1}} dv = \int{\frac{1}{x}}} dx \displaystyle{\Longrightarrow \log(\log(v) - 1) = \log(x) + C \Longrightarrow \log(v) - 1 = C x \Longrightarrow v = \frac{y^{\prime}}{x}...
  3. 5

    Second order non-linear differential equation involving log

    thanks for the help. I don't know how to evaluate \frac{d \log(u)}{d \log(x)} I understand that I would be differentiating \log(u) as a function of \log(x), but I don't see how that is a function of \log(x).
  4. 5

    Second order non-linear differential equation involving log

    thanks for the help. OK, that was productive. Here's what I did: v := \frac{y^{\prime}}{x} \Longrightarrow v^{\prime} = \frac{x y^{\prime \prime} - y^{\prime}}{x^2} \Longrightarrow x^2 v^{\prime} + y^{\prime} = x y^{\prime \prime} Substituting, x^2 v^{\prime} + y^{\prime} =...
  5. 5

    Second order non-linear differential equation involving log

    EDIT: my problem is solved, thank you to those who helped Homework Statement Solve: x y^{\prime \prime} = y^{\prime} \log (\frac{y^{\prime}}{x}) Note: This is the first part of an undergraduate applications course in differential equations. We were taught to solve second order...
  6. 5

    Compute the flux from left to right across a curve

    Gotcha; that makes a bit more sense now. Thanks for the generalization!
  7. 5

    Compute the flux from left to right across a curve

    Thanks for the reply. Alright... I think I need to work on my intuition for this stuff to better understand your help. Anyway, I think I've got something decent, and I'm done with it. For anyone looking at this thread for help, I found this link helpful...
  8. 5

    Compute the flux from left to right across a curve

    Thanks for the response. "If you sketch your curve and thing of things flowing left to right, they would be flowing in the direction of increasing r from your curve." I don't understand what you are saying here. Could you please explain more explicitly? I am thinking you are talking about...
  9. 5

    Compute the flux from left to right across a curve

    Homework Statement Compute the flux of \overrightarrow{F}(x,y) = (-y,x) from left to right across the curve that is the image of the path \overrightarrow{\gamma} : [0, \pi /2] \rightarrow \mathbb{R}^2, t \mapsto (t\cos(t), t\sin(t)). A (2-space) graph was actually given, and the problem...
  10. 5

    Prove that ℝ has no subspaces except ℝ and {0}.

    Assuming we are dealing in the realm of undergraduate linear algebra... Continuing on what WannabeNewton said, think about why \mathbb{R} is a subspace of itself, and then consider some nonempty set that is not \mathbb{R} or the set consisting of just 0. Recall that a nonempty set is a...
  11. 5

    Just a question about vertical and horizontal line tests

    The horizontal and vertical line test is an intuitive way to see if a graph belongs to an injective function. To be clear, if the graph belongs to a function, and a horizontal line drawn anywhere on the graph means that the line will only intersect the graph at most once, then the function is...
  12. 5

    Use Lagrange multipliers to find the eigenvalues and eigenvectors of a matrix

    thanks for the reply. I still find the concept odd, but it is a bit more clear, thanks. I think I just need time for this to settle in my mind, maybe. A few questions, though. One, when we are referring to the eigenvectors as e_1 and e_2, are we referring to them as [1,0] and [0,1], the...
  13. 5

    Function question. Is this correct?

    Do you mean letting y=5? If so, then yes, you get \pm \sqrt{-1}, which is not a real number, and therefore there does not exist any x in the domain of f such that f(x)=5. Consequently, the range of f does not include 5, for example. That is enough to show that surjectivity is not held by f, as...
  14. 5

    Function question. Is this correct?

    What is your reasoning? Substitute some values for x into f and see what you get, or what you can't get. Does your answer change? Even better, use graphing software to visualize f, and then everything should be pretty clear. (Example, search "wolframalpha" into a search engine and type "f(x) = 4...
  15. 5

    Function question. Is this correct?

    Your notation for the function definition isn't correct, I think. I believe you meant f : \mathbb{R} \rightarrow \mathbb{R}, x \in \mathbb{R} \mapsto 4 - x^2. Here, \mathbb{R} is both the domain and codomain of f. Surjectivity is the property that the image of the domain of f, which is...
  16. 5

    Use Lagrange multipliers to find the eigenvalues and eigenvectors of a matrix

    Homework Statement Use Lagrange multipliers to find the eigenvalues and eigenvectors of the matrix A=\begin{bmatrix}2 & 4\\4 & 8\end{bmatrix} Homework Equations ... The Attempt at a Solution The book deals with this as an exercise. From what I understand, it says to consider...
  17. 5

    Does this epsilon delta limit proof check out?

    Alright, after probably too much work, I think I've gotten it worked out with delta-epsilon proof. The following is the tactic I used. Just to give clarity, we want to show that, if, for any real number \varepsilon > 0, there exists a real number \delta > 0 such that 0 < ||(x,y) - (0, 0)||...
  18. 5

    Does this epsilon delta limit proof check out?

    You might have made a common mistake, but I might be wrong. You want to find a \delta so that you can get from 0 < x^2 + y^2 < \delta (*) to | \frac{x^2 \sin^2(y)}{x^2 + 2y^2} - 0| < \epsilon (**) by good old, cold, mathematical logic and stuff (it would be good to state beforehand that your...
  19. 5

    Need to prove if x<1 and m>=n then x^m<=x^n

    haha, yeah, you are correct. I've been meaning to fix that, :} EDIT: I don't seem to have the ability to edit that reply anymore, :S. Oh well...
  20. 5

    Contrapositive of a (if p and q, then r) statement?

    If I understand correctly, since our conclusion is a "result 1 OR result 2" conclusion, and it appears we can only have one or the other, but not both, i.e. the "OR" as an "exclusive-or," then we must show both that result 1 can occur while result 2 does not occur, and, likewise, that result 2...
  21. 5

    Chain Rule Trig Derivative Problem

    haha, yeah, definitely. Yeah, I remember finding the chain rule a little weird. Just sit on it for a bit, and don't worry if it doesn't make "perfect sense" immediately.
  22. 5

    Need to prove if x<1 and m>=n then x^m<=x^n

    Alright, so, I gather the theorem is, "For any x \in \mathbb{R}, \left\{m, n\right\} \subseteq \mathbb{N}, if both 0 \leq x \leq 1 and m \geq n, then x^m \leq x^n." I'd start by breaking it into cases. First, consider when x=0. Then, clearly we have our conclusion. Similar, if we consider...
  23. 5

    Contrapositive of a (if p and q, then r) statement?

    I believe the contrapositive here would be what you guessed it to be. As others have said, I think its easier to prove directly. For fun, let's prove the contrapositive, worded better for proving: "For any vector u, in some vector space V, and any scalars a,b, in some field F, consider the...
  24. 5

    Chain Rule Trig Derivative Problem

    A good way to do this is to break it down so its easier to apply the chain rule for derivation. I'll assume we are considering the function f(x)=\sin((\pi x)^2). To break this down, define the function S(x)=x^2. Also, define the function g(x)=\pi x. (Note that these x parameters in...
  25. 5

    Proving a floor function

    I'll assume were working with x \in \mathbb{R}. Just to be clear of the definition of the floor function, it is the function \left\lfloor \right\rfloor : \mathbb{R} \longrightarrow \mathbb{Z}, with the mapping x \in \mathbb{R} \longmapsto \mathrm{max} \left\{ y \in \mathbb{Z} : y \leq x...
  26. 5

    Expressing geometrically the nth roots of a complex number on a circle

    Yeah, that's what I am thinking. Thanks for the reply
  27. 5

    Expressing geometrically the nth roots of a complex number on a circle

    Homework Statement Let z \in \mathbb{C}. Prove that z^{1/n} can be expressed geometrically as n equally spaced points on the circle x^2 + y^2 = |z|^2, where |z|=|a+bi|=\sqrt{a^2 + b^2}, the modulus of z. Homework Equations // The Attempt at a Solution My problem is that I am...
  28. 5

    Third degree Taylor polynomial in two variables

    Thanks for the reply, HallsofIvy. Oh dear, I see. Hopefully this will be my last change of answer: f(x,y) = (1 - \frac{1}{2}x^2) (1 - xy) = 1 - xy - \frac{1}{2}x^2 Do I now have the correct idea? D=
  29. 5

    Third degree Taylor polynomial in two variables

    Thanks for the reply, Zondrina. Ohhh, so that is what is meant by "n-th degree!" Alright, then I would have f(x,y) = (1 - \frac{1}{2}x^2) (1 - xy + x^2y^2 - x^3y^3) so f(x,y) = 1 - xy + x^2y^2 -x^3y^3 - \frac{1}{2}x^2 + \frac{1}{2}x^3y - \frac{1}{2}x^4y^2 + \frac{1}{2}x^5y^3...
  30. 5

    Third degree Taylor polynomial in two variables

    Homework Statement Find the third degree Taylor polynomial about the origin of f(x,y) = \frac{\cos(x)}{1+xy} Homework Equations The Attempt at a Solution From my ventures on the Internet, this is my attempt: I see that \cos(x) = 1 - \frac{1}{2}x^2 + \frac{1}{4!}x^4 - \cdots...
Top