Results 1 to 4 of 4

Math Help - A tricky ODE and finding extrema questions!

  1. #1
    Newbie
    Joined
    Aug 2008
    Posts
    2

    A tricky ODE and finding extrema questions!

    Hi,

    my name is Dimos and I am coming from Greece.

    Right to the point.

    1) Can somebody help me with the:

     (2x*cos(y)+y*cos(x))+(-x^2*sin(y)+x)*dy/dx=0

    ODE?

    I ve tried to find a integration factor and change variable (e.g. v=2x*cos(y) ). I have no clue! Is this ODE a suitable example for polytechnics student?

    2) Is there any higher order derivative test to find extrema of multi variable functions? Where can i find more details?

    3) Can the use of Lagrange multipliers reveal the nature of extrema (i.e. is it a local minimum, local maximum or saddle point)? How?

    Thank you in advance for any help!

    Dimos
    Athens.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Joined
    Aug 2007
    From
    USA
    Posts
    3,111
    Thanks
    2
    Just a reality check. It is SO CLOSE to "exact" that I am first inclined to make sure there is not just a typo.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Aug 2008
    Posts
    2
    Quote Originally Posted by TKHunny View Post
    Just a reality check. It is SO CLOSE to "exact" that I am first inclined to make sure there is not just a typo.
    I thought it was a typo. If x in the dy/dx term was sin(x), the ODE would be exact and having the solution

    x^2*cos(y)+y*sin(x)=c

    the person gave it to me for solving is not available at the moment. So I assumed it s correct.

    Anyway it may be an interesting (and hard to solve) ODE.

    Thanks for your help TKHunny!

    Dimos
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Rhymes with Orange Chris L T521's Avatar
    Joined
    May 2008
    From
    Chicago, IL
    Posts
    2,844
    Thanks
    3
    Quote Originally Posted by ouranosky View Post
    2) Is there any higher order derivative test to find extrema of multi variable functions? Where can i find more details?
    Do you know about the second partials test?

    THEOREM: Second Partials Test.

    Let f be a function of two variables with continuous second order partial derivatives in some disk centered at a critical point (x_0,y_0), and let

    D=f_{xx}(x_0,y_0)f_{yy}(x_0,y_0)-f_{xy}^2

    (a) If D>0 and f_{xx}(x_0,y_0)>0, then f has a relative minimum at (x_0,y_0)

    (b) If D>0 and f_{xx}(x_0,y_0)<0, then f has a relative maximum at (x_0,y_0)

    (c) If D<0 , then f has a saddle point at (x_0,y_0)

    (d) If D=0, then no conclusion can be drawn
    Source: Calculus : Multivariable by Anton, Bivens, Davis. 8th Ed.

    3) Can the use of Lagrange multipliers reveal the nature of extrema (i.e. is it a local minimum, local maximum or saddle point)? How?
    Yes you can show these things. I don't have the time to go through the process, but you need to use this equation(s):

    \nabla f(x_0,y_0)=\lambda\nabla g(x_0,y_0) <---Use when you have two variables and one constraint.

    \nabla f(x_0,y_0,z_0)=\lambda\nabla g(x_0,y_0,z_0) <---Use when you have three variables and one constraint.

    where \lambda is the Lagrange Multiplier.

    I hope this helps!

    --Chris
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. [SOLVED] Tricky Applied Extrema Problem
    Posted in the Calculus Forum
    Replies: 4
    Last Post: July 14th 2010, 09:22 AM
  2. finding absolute extrema
    Posted in the Calculus Forum
    Replies: 1
    Last Post: March 3rd 2010, 03:37 AM
  3. Finding Extrema
    Posted in the Calculus Forum
    Replies: 3
    Last Post: November 22nd 2009, 09:49 AM
  4. finding the absolute extrema
    Posted in the Calculus Forum
    Replies: 1
    Last Post: April 27th 2009, 02:16 PM
  5. First derivative, relative extrema questions
    Posted in the Calculus Forum
    Replies: 4
    Last Post: July 17th 2008, 12:07 AM

Search Tags


/mathhelpforum @mathhelpforum