System of differential equation with limit

Question

I am given the relation $\dfrac{dy(x)}{dx}=az(x)$ and $\dfrac{dz(x)}{dx}=bz(x)$ (with $a$, $b$ are constants ), and after solving I have found $\lim\limits_{x\to\infty}\dfrac{y(x)}{z(x)}=\dfrac{a}{b}$, then what can we say about $y(x)$ and $z(x)$?

Answer 1

Starting with $\dfrac{dz(x)}{dx}=bz(x)$ we can actually solve directly for $z$. Start by rearranging the equation $\dfrac{dz}{z}=bdx$ and then integrate both sides, $\ln(z)=bx+c_0$ or $z=c_0e^{bx}$.

Plugging this in to the other relation we obtain $\dfrac{dy(x)}{dx}=ac_0e^{bx}$ and $dy(x)=c_0ae^{bx}dx$. And taking integrals $y=\frac{c_0a}{b}e^{bx}+c_1$.

As for your original statement of the limit we can see that $$\lim_{x \to \infty}\dfrac{y(x)}{z(x)}=\dfrac{\frac{c_0a}{b}e^{bx}+c_1}{c_0e^{bx}} \to \dfrac{\frac{c_0a}{b}e^{bx}}{c_0e^{bx}}=\frac{a}{b}$$