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First Order Differential Equation

  1. Jun 1, 2010 #1
    1. The problem statement, all variables and given/known data
    Find the general solution for: x y' - 2y = x +1 (x>0)

    2. Relevant equations

    3. The attempt at a solution
    I have literally no idea how to start this. I've tried seperating variables but ended up with:

    [tex]\frac{y'-1}{2y+1}[/tex] = [tex]\frac{1}{x}[/tex] but that isn't solvable due to the y'-1 (at least I don't know how if it is). Any help is greatly appreciated!

    Edit: Ok, so I've used integrating factors. If you take the integrating factor to be x-2 (i can show how if needed) and then use that, I find:

    [tex]\frac{d}{dx}(x^{-2}y) = x^{-2}+ x^{-3}[/tex] so the equation integrates to:

    [tex] y = cx^{2}-x-\frac{1}{2}[/tex] does that seem about right? We have disagreement in our group :/
    Last edited: Jun 1, 2010
  2. jcsd
  3. Jun 1, 2010 #2


    Staff: Mentor

    An integrating factor will work.

    Rewrite the DE as y' - (2/x)y = 1 + 1/x

    The integrating factor is
    [tex]e^{\int \frac{-2}{x}dx}[/tex]
  4. Jun 1, 2010 #3
    Thanks! Just edited above the work I managed with that! Does that seem right?

    Ok, rather than start a new thread (which I can do if it would be easier), I thought I'd put in the new question I have here:

    1. The problem statement, all variables and given/known data
    i) Find the general solution for y'' + 2y' + 3y =0
    ii) Give the basis for the vector space of all solutions of this equation.
    iii) Describe the behaviour of all solutions as t tends to infinity
    iv) If this equation models a mass-spring system with no external force, state whether the motion is underdamped, critically damped or overdamped.

    2. Relevant equations

    3. The attempt at a solution
    i) Easy enough:
    y(t)=e-t(([tex]Ccos(\sqrt{2}t) + Dsin(\sqrt{2}t)[/tex]))
    ii) I don't really understand this question, but from what I can gather after brief research:
    Basis{(e-t[tex]cos\sqrt{2}t[/tex]),(e-t[tex]sin(\sqrt{2}t)[/tex]))}. Can someone explain this question please?
    iii) as t tends to infinity, y(t) tends to 0, due to the exponential decay.
    iv) I'm not really sure here. All my notes on mass-spring systems don't really cover differential solutions when I haven't worked from a frequency point-of-view. I think it shows an underdamped system.

    Are those about right?
    Last edited: Jun 1, 2010
  5. Jun 1, 2010 #4


    Staff: Mentor

    Yes, that's the basis. The solution space is a vector (or function) space that is spanned by those two functions. Every solution of the given DE is a linear combination of those two functions.
    Yes. For iv, over-, under-, and critically damped relate directly to the discriminant in the solution of the characteristic equation (which in this case is r2 + 2r + 3 = 0).

    The discriminant D is b4 - 4ac. If D < 0, the system is underdamped, since there is still a fair amount of oscillation. If D > 0, there is no oscillation, and the system is overdamped. If D = 0, the system is critically damped.
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