1. Limited time only! Sign up for a free 30min personal 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!

Homework Help: Second order differential equation using substitution

  1. Mar 29, 2012 #1
    1. The problem statement, all variables and given/known data

    2. Relevant equations
    Use the substitution [itex]x=\cos\theta[/itex]

    3. The attempt at a solution
    I started off by listing:




    But don't know whether this helps, or where to go next. Could someone please give me a hint at how to approach this, I'd prefer not to have a full solution, but really am desperate for a starting point!

    With very many thanks,

  2. jcsd
  3. Mar 29, 2012 #2


    User Avatar
    Science Advisor

    Since you are differentiating with respect to [itex]\theta[/itex], you need to use the chain rule:
    [tex]\frac{dy}{d\theta}= \frac{dy}{dx}\frac{dx}{d\theta}= -sin(\theta)\frac{dy}{dx}[/tex]
    [tex]= \frac{d}{d\theta}(\frac{dy}{d\theta})[/tex]
    [tex]= \frac{d}{d\theta}(cos(\theta)\frac{dy}{dx}[/tex]
    [tex]= -sin(\theta)\frac{dy}{dx}+ cos(\theta)\frac{d}{d\theta}\frac{dy}{dx}[/tex]
    [tex]= -sin(\theta)\frac{dy}{dx}+ cos^2(\theta)\frac{d^2y}{dx^2}[/tex]
    Last edited by a moderator: Mar 29, 2012
  4. Mar 31, 2012 #3
    Thanks very much! Have got it now!
  5. Apr 7, 2012 #4
    Hi! I was wondering why
    d/dθ (dy/dθ ) does not =d/dθ (-sin(θ) dy/dx)?

    Also, once you have the relevant expressions how do you solve the differential equation? Do you have to form the auxilliary equation? I tried to do that but it doesn't work out nicely.

Share this great discussion with others via Reddit, Google+, Twitter, or Facebook