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Second Order Equation  Change of variables 
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#1
Feb2813, 10:56 AM

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 


#2
Feb2813, 02:20 PM

P: 298

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



#3
Mar113, 04:46 AM

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 


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