Consider the differential equation $y^{\prime }\left(x\right)=(y+2)(y-3).$ In which regions are the solutions increasing?

For the autonomous differential equation $y^{\prime }\left(t\right)=3y-6$, find the equilibrium solution.

Sketch the solution curve that corresponds to the initial condition $y\left(0\right)=1$ for the differential equation $y^{\prime }\left(t\right)=8-4y$.

Solve the differential equation: $y^{\prime }\left(t\right)-4y=3$

Solve the Bernoulli equation: $z^{\prime }\left(t\right)-4z=8z^2$ for $z\left(t\right)$.

Suppose a car loan is described by the differential equation $C^{\prime }\left(t\right)=0.05C-400$, where $C\left(t\right)$ is the unpaid balance and $C\left(0\right)$ is the initial loan amount. What is the largest amount you can borrow without the balance increasing each month?

In a financial model, the amount $A$ grows according to $\frac{dA}{dt}=h\left(A\right)$. What role does the function $h\left(A\right)$ play in determining $A\left(t\right)$?

A fish population in a pond starts at $50$ at $t=0$. After $10$ days, the population is $200$. The carrying capacity is $3200$. After how many days will the population reach half the carrying capacity? Round your answer to nearest whole number.

Solve the initial value problem: $x^{\prime \prime }\left(s\right)=s{e}^s$, $x\left(0\right)=1$, $x^{\prime }\left(0\right)=0$.