QUESTION $1(15$ MARKS$)$

a$)$ Find the solution to the Bernoulli equation

$x{y}^2\frac{dy}{dx}+y^3=1.,n{y}^2$

$\frac{dy}{dx}\frac{y}{x}=n{y}^{-2}$

b$)$ The population of fish in a pond satisfies the differential equation

$\frac{dp}{dt}=360-3p$

If there are initially $10$ fish in the pond and $t$ is measured in years,

i$)$ find the number of the fish at any time $t,p(t).$

$(4$ marks$)$

ii$)$ how long will it take the number of fish to reach $102?$

$(2$ marks$)$

iii$)$ what is the number of fish after $12$ months?

$(2$ marks$)$