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Differential equation formation

  1. Feb 4, 2014 #1
    1. The problem statement, all variables and given/known data
    If α is an arbitrary constant and a a fixed constant show that
    xcos α + ysin α = a

    is the complete primitive of the equation

    (y - xdy/dx)^2 = a^2( 1 + (dy/dx)^2)


    2. Relevant equations



    3. The attempt at a solution

    FIrst I found the first derivative by differentiating implicitly,

    cosα + (dy/dx)sinα = 0

    But now I'm looking for something to sub to try and get the differential equation and eliminate the arbitrary constants, but I keep getting into a loop and cancelling everything basically...

    Please help.
     
  2. jcsd
  3. Feb 5, 2014 #2
    Hi lionely!

    You have got two equations. Solve for cos(α) and sin(α) and plug them in the expression ##\cos^2\alpha + \sin^2\alpha=1##.
     
  4. Feb 5, 2014 #3
    But if I use those equations I have all I get is cos(α) in terms of x,y and sin(α) or dy/dx. SO when I put it in the identity I still can't get rid of the arbitrary constants.
     
  5. Feb 5, 2014 #4

    pasmith

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    Homework Helper

    You goal is to confirm that the left hand side of the ODE is equal to the right hand side. If you work out [itex]y[/itex] and [itex]dy/dx[/itex] and just substitute them into the ODE then everything cancelling is exactly what you want!
     
  6. Feb 5, 2014 #5
    But I want to move from the primitive to the differential.
     
  7. Feb 5, 2014 #6

    pasmith

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    Homework Helper

    Starting from the given solution and trying to manipulate it to obtain the given ODE is not going to work. Your solution is linear; the only way to eliminate the constants of integration is to differentiate twice to obtain [itex]y'' = 0[/itex]. That doesn't help you.

    There are two approaches you can take:
    • Solve the ODE and show that its general solution takes the given form (this can be done; the easiest way is to differentiate both sides with respect to [itex]x[/itex]; unfortunately the resulting 2nd order equation admits solutions which don't satisfy the original first order ODE, and these solutions must be identified and eliminated).
    • Substitute the given form into the ODE and show that everything cancels (this is straightforward).

    Actually, I suppose that if you're asked to show that a primitive is a complete primitive then you should use the first approach. Otherwise you've just shown that it is a primitive.
     
  8. Feb 5, 2014 #7
    If I can't move from the primitive to the differential equation, how the heck did the author of the book get it? This thing is just puzzling me.
     
  9. Feb 6, 2014 #8
    Let ##\cos\alpha=c## and ##\sin\alpha=e##, then you have the following system of linear equations:

    $$cx+ey=a$$
    $$c+(dy/dx)e=0$$

    Solve the above equations for ##c## and ##d##.
     
  10. Feb 6, 2014 #9
    C = (a-ey)/x

    and e= -c/(dy/dx) ? I guess.. Then put these in sin^2 a + cos^2a = 1?
     
  11. Feb 6, 2014 #10
    I meant that find ##c## and ##e## in terms of x,y and dy/dx and then plug them in the identity.
     
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