Find the general solution to the separable differential equation $3t\frac{dw}{dt}=6t^2$.

A rumor spreads in a school according to $\frac{dS}{dt}=aS(1-\frac{S}{B})$, where $a$ is the spread rate, $B$ is the total number of students, and $S_0$ is the initial number of students who know the rumor. Find $S\left(t\right)$ in terms of $a$, $B$, $S_0$.

A $600$-$\text{L}$ tank is initially filled with pure water. A potassium nitrate solution with a concentration of $15\text{ g/L}$ flows into the tank at a rate of $5\text{ L/min}$. The thoroughly mixed solution is drained from the tank at a rate of $5\text{ L/min}$. Let $m\left(t\right)$ be the mass of potassium nitrate (in grams) at time $t$ (in minutes). Determine the initial value problem that models the mass of potassium nitrate in the tank.

A cup of tea is initially at a temperature of ${85}^{\circ }\text{C}$ and cools in a room maintained at ${20}^{\circ }\text{C}$. If the cooling constant is $k=0.03$ ${\text{min}}^{-1}$, after how many minutes will the tea reach ${40}^{\circ }\text{C}$? Round your answer to the nearest whole number.

Consider the predator-prey system given by the equations: $x^{\prime }\left(t\right)=-2x+5xy$, $y^{\prime }\left(t\right)=y-3xy$. Determine the equilibrium points for this system.