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Second Order Equation - Change of variables

  1. Feb 28, 2013 #1
    Hello there,

    I am facing the second order ODE in the unknown function $$y(t)$$
    $$ \ddot{y} = a \dot{y} y - b \dot{l} l - c\dot{l} + d$$ $$a, b, c, d$$ positive constants, such that $$ \frac{a}{b} = \frac{d}{c}$$

    I would like to understand more about it before relying on numerical methods.

    So far, I have only determined that the long term solution as $$t \to 0$$ is a linear function, $$y(t) = \frac{a}{b}t$$.
    If the boundary condition on $$\dot{y}$$ is zero, I get interesting transient behaviour. If the boundary condition on $$\dot{y}$$ equals $$ \frac{a}{b}$$ i get, as expected, a linear solution, which though starts getting very wiggly and wildly oscillating (stiff equation?).

    Is there some change of variable, some tecnique, some trick I could use to reduce my equation in a form friendlier for analytic investigations?

    Many thanks
  2. jcsd
  3. Feb 28, 2013 #2
    what is l in the ode for y(t)? Is it a second dependent variable as in l(t)
  4. Mar 1, 2013 #3
    I made a mistake in writing the equation down, sorrry for the confusion created.

    The ODE in the unknown $$y(t)$$ looks like

    $$\ddot{y}= a \dot{y} t - b y \dot{y} -c \dot{y} + d$$
    with $$ \frac{a}{b} = \frac{d}{c}$$

    (the costants ratios allow the linear solution $$y (t) = \frac {a}{b} t$$, B.C. allowing)

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