Second Order Equation - Change of variables

[muzialis]

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

 

[bigfooted]

what is l in the ode for y(t)? Is it a second dependent variable as in l(t)

 

[muzialis]

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)

Thanks