Let's begin:
7. We want to solve the differential equation \[ \frac{d y}{d x}=\frac{x^{2}}{y} \]. We can start by multiplying both sides by \(y\) to simplify, which gives us \(y \frac{dy}{dx} = x^2\). Now it's clear we have a separable differential equation, so we can divide by \(y\) and multiply by \(dx\) on both sides: \[y dx = x^2 dx\]. Next, we integrate both sides: \[ \int y dy = \int x^2 dx\]. The left integral results in \(\frac{1}{2}y^2\) and the right integral results in \(\frac{1}{3}x^3 + c\) where \(c\) is the constant of integration. Thus, the general solution will be \[y = \sqrt{\frac{2}{3}x^{3} + 2c}\].
8. We want to solve the differential equation \[ \frac{d y}{d x}=x^{2} y \]. This is a first order linear non-homogeneous differential equation which can be solved using a method called integrating factor. We define integrating factor to be \( e^{\int{P(x)dx}}\), where \(P(x)\) is the coefficient of \(y\) in the given differential equation which in the current problem is \(x^2\). By calculating it, we get \( e^{\int{x^2}dx} = e^{\frac{x^3}{3}} \). Next, we multiply the given differential equation by the integrating factor and integrate the result: \[\int{ e^{\frac{x^3}{3}} ( x^2 y) dx = \int{ e^{\frac{x^3}{3}} dx}\]. Simplifying this gives us the solution \[ y = e^{-\frac{x^3}{3}} \int{ e^{\frac{x^3}{3}} dx + c}\] where \(c\) is the constant of integration.
9. The final problem is about the mathematical modelling of a hypothetical population but since the question is incomplete, I am afraid we cannot provide a solution without having sufficient information such as the details of the population growth rate, constraints, and the specific modelling system to be used. For instance, we may apply models like exponential growth, logistic growth, etc., but these specifics are not given.