AP Calculus BC Unit 7: Differential Equations

A slope field for $\frac{dy}{dx}=x-y$ shows slopes of $0$ along the line:

Which of the following differential equations is NOT separable?

Solve the separable ODE $\frac{dy}{dx}=3x^{2}y$ with initial condition $y(0)=2$.

Verify that $y=C{e}^{2x}+3$ is the general solution to $\frac{dy}{dx}=2(y-3)$. What is the particular solution satisfying $y(0)=7$?

Solve the separable ODE $\frac{dy}{dx}=\frac{x}{y}$ with $y(0)=3$.

Use Euler's method with step size $h=0.5$ to approximate $y(1)$ for $\frac{dy}{dx}=x+y$ with $y(0)=1$.

The solution to a logistic equation passes through $(0,100)$ and approaches $L=800$ as $t\to \infty$. Using $P=\frac{L}{1+A{e}^{-rt}}$ with $r=0.5$, find $A$.

For the ODE $\frac{dy}{dx}=\frac{y^2}{x}$, find the general solution.

For the logistic model $dP/dt=0.2P(1-P/500)$ with $P(0)=50$, at what population is the growth rate $dP/dt$ maximized?

The logistic growth model is $\frac{dP}{dt}=0.3P\left(1-\frac{P}{1000}\right)$. As $t\to \infty$, what does $P$ approach?

For $\frac{dy}{dx}=ky$ with $k<0$, the general solution is $y=C{e}^{kx}$. This represents exponential decay. If $y(0)=10$ and $y(3)=5$, find $k$.

A population grows according to $P^{\prime }=0.04P$. If $P(0)=500$, find $P(10)$.