1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Analysis Question 2

  1. Apr 28, 2009 #1
    1. The problem statement, all variables and given/known data
    Suppose f:R→R and g:R→R are both differentiable and that f'(x)=g(x) and g'(x)=-f(x) for all x ∈ R; f(0)=0 and g(0)=1.
    Prove : (f(x))²+(g(x))²=1 for all x ∈ R.


    2. Relevant equations



    3. The attempt at a solution
    I know I need the find d/dx[f(x)²+g(x)²]=d/dx[1], but I am not sure what that is going to help me find and how to use the result.
     
  2. jcsd
  3. Apr 29, 2009 #2
    Proceed with d/dx[f(x)²+g(x)²]=d/dx[1] and use the fact that f'(x)=g(x) and g'(x)=-f(x) for all x ∈ R to get an equality that is clearly true. Then see if you can make all of your steps reversible.
     
  4. Apr 29, 2009 #3
    Prove that both f and g are twice differentiable.

    Obtain the differential equations: f'' = -f and g'' = -g

    Then you know that the function sinx, with initial conditions sin(0) = 0 and d(sin0)/dx = 1 and the funtion cosx with initial condition cos(0) = 1 and d(cos0)/dx = 0 are the solutions to the differential equations.

    If you aren't familiar with defining cosine and sine with a differential equation, then just do what snipez advised. Assume that
    d(f^2(x) + g^2(x))/dx = d(1)/dx until you get an equality that is true and try to reverse it. Also remember that if y' = 0 then y = c, where c is some real number.
     
    Last edited: Apr 29, 2009
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Analysis Question 2
  1. Analysis question (Replies: 0)

  2. Analysis question (Replies: 1)

  3. 2 analysis question (Replies: 3)

  4. An analysis question (Replies: 20)

  5. An analysis question (Replies: 4)

Loading...