-1

I' trying to solve this problem but would need some help:

Let $N=N(t)$ denote the size of a certain population, $X=x(t)$ the total product and $y(t)=X(t)/N(t)$ the product per capita at time $t$. Suppose: $$\frac {\dot N}{N}= \alpha-\beta \frac{N}{X}, X=N^{ \sigma }, \sigma \ne1 $$

Show that the above system can write as a differential equation of y in form of $\dot y=ay(t)+b$.

M47145
  • 4,106
ocram
  • 743
  • 2
  • 7
  • 15

1 Answers1

1

Since $y=\frac{X}{N}$, it follows that $\frac{\dot{y}}{y} = \frac{\dot{X}}{X} - \frac{\dot{N}}{N}$ (Take logs of both sides then differentiate wrt time). Call this equation (1).

Also, since $X=N^{\sigma}$, it also follows that $\frac{\dot{X}}{X} = \sigma\frac{\dot{N}}{N}$ If we substiute this into equation (1) we get $\frac{\dot{y}}{y} = (\sigma-1)\frac{\dot{N}}{N}$. Call this equation (2).

Substituting equation (2) into the main equation gives you your answer:

$ \frac{\dot{N}}{N} = \alpha -\frac{\beta}{y} $

$\frac{1}{\sigma-1}\frac{\dot{y}}{y} = \alpha -\frac{\beta}{y} $

$\dot{y} = \alpha(\sigma-1)y -\beta(\sigma-1) $

Greg
  • 682