# Second Order Equation - Change of variables

by muzialis
Tags: equation, order, variables
 P: 152 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
 P: 152 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