Differential equations + exponenetial models

7a

A differential equation is given by $\displaystyle \frac{dx}{dt}=-kte^\frac{x}{2}$, where k is a positive constant. Note: The power to e is x/2, tried making it bigger but still suck at using latex.

(i) Solve the differential equation.

(ii) Hence, given that x=6 when t=0, show that $\displaystyle x=-2\ln(\frac{kt^2}{4}+\frac{1}{e^3})$.

7b The population of a colony of insects is decreasing according to the model $\displaystyle \frac{dx}{dt}=-kte^\frac{x}{2}$, where x thousands is the number of insects in the colony after time t minutes. Initially, there were 6000 insects in the colony.

Given that k=0.004 , find:

(i) the population of the colony after 10 minutes, giving your answer to the nearest hundred.

(ii) the time after which there will be no insects left in the colony, giving your answer to the nearest 0.1 of a minute.