# [Numerical] System of first order ordinary diff equations with given asymptotic

by ala
Tags: numerical diff
 P: 813 Okay, bellow is my attempt: Let: $$y(x)=c(1-w(x)exp(-kx))$$ Then $$w(x)=(1-y/c)exp(kt)$$ $$\dot{w}=-\frac{\dot{y}}{c}exp(-kx)+k \ (1-y/c)exp(-kx)$$ Now time to make some substitutions $$\dot{w}=-\frac{f(y)}{c}exp(kt)+k \ (1-\left(c(1-w \ exp(-kx)) \right)/c)exp(kt)$$ $$=-\frac{f(y)}{c}exp(kx)-w \ \left(1-\frac{k}{c}\right)exp(kx)+k \ w \$$ where $$y$$ is given above as: $$y=c (1- w \ exp(-kx))$$ and $$f(y)$$ is the original differential equation. edit: The above only seems useful if $$\frac{1}{x}$$ is much bigger then $$k$$.