The population of rabbits in a farm satisfies the logistic growth model $\frac{\mathrm{d} N}{\mathrm{~d} t}=5 N\left(8-\frac{N}{10}\right)$, where $t$ is measured in hours and the initial population is 30 .
i) What is the carrying capacity of the population? Answer:
数字
ii) What will the population’s size be when the population is growing the fastest?
$N=50$
$N=40$
$N=30$
$N=60$

Find the value of $a$ such that the function
$$
y=a x^2-7
$$
satisfies the differential equation
$$
x y^{\prime}-y+3(x-3)^2=34-18 x
$$
$$
a=\text { 数字 }
$$

The ordinary differential equation
$$
y^{\prime \prime}=y^{\prime} t^2+3 y-2
$$
is linear.
TRUE
FALSE

Find the general solutions of:
$$
\frac{d y}{d x}+2 x y=6 x
$$
Write your answer in the form $y=$ expression. Use ” $C^{\prime \prime}$ as your constant of integration. If necessary use $\exp \left(z^{\wedge} 2\right)$ for $e^{z^2}$.