Question:

a) Solve \(\frac{dy}{dx}=2x{{y}^{2}}\), with \(y\left( 0 \right)=1\). What is when reaches infinity?

b) Find the general solution to

c) Use Euler’s Method with a step size of 0.5 to estimate y(2) if \(y'=xy-1\), where \(y\left( 1 \right)=2\)

d) A population of insects in a region will grow at a rate that is proportional to their current population. In the absence of the outside factors, the population will quadruple in two weeks time. On any given day there is a net migration into the area of 20 insects and 15 insects are eaten by a local bird and 10 insects die of natural causes. If there are initially 150 insects in the area, will the population survive? If not, when do they die out?