Exercise $3($Population model of von Bertalanffy$)$

Usually the growth rate of a specie is expressed via the mass of carbon it produces The von Bertalanffly model has the following form:

$\frac{d}{dt}m(x,t)=a{x}^2(t)-b{x}^3(t)$

where $m$ is the biomass, $x$ a characteristic size of the specie, $$ the rate of growth of the biomass and $b$ the rate of consumption of the biomass.

$($a$)$ Assuming that $m=x^3$ write the model as a differential equation of the variable $x(t).$

$($b$)$ Write the dimensions of the constants$/$parameters $,b$ and $.$

$($c$)$ Compute the matrix of the dimensions $A.$

$($d$)$ Define the dimensionless problem using proper characteristic variables for the size and the time, for example: $\frac{}{b}$ and $\frac{}{b}$ respectively.

$($e$)$ If