AP Calculus AB Unit 7: Differential Equations

A slope field for a differential equation shows short line segments with slope $x+y$ at each point $(x,y)$. Which differential equation corresponds to this slope field?

For the initial value problem $\frac{dy}{dx}=x-y$, $y(0)=1$, use Euler's method with step size $h=1$ to estimate $y(2)$.

Which of the following is the solution to $\frac{dy}{dx}=2x$ with initial condition $y(0)=3$?

Solve $\frac{dy}{dx}=-2x{y}^2$ with $y(0)=1$.

A slope field for $\frac{dy}{dx}=\frac{x}{y}$ is sketched. Which statement best describes a solution curve passing through $(0,3)$?

Solve the separable ODE $\frac{dy}{dx}=\frac{x}{y}$ with $y(0)=4$. (Give the solution for $y>0$.)

Radioactive carbon-14 decays according to $\frac{dN}{dt}=-0.000121N$. If $N_0=1000$ atoms initially, how many atoms remain after $5730$ years (one half-life)?

The general solution to $\frac{dy}{dx}=ky$ is $y=C{e}^{kx}$. If $y(0)=5$ and $y(2)=20$, find $k$.

A bacteria culture doubles every 3 hours. If the initial count is $N_0$, what is the count after $t$ hours?

On a slope field for $\frac{dy}{dx}=y-x$, the slopes along the line $y=x$ are:

Newton's law of cooling gives $\frac{dT}{dt}=-k(T-T_s)$, where $T_s=20$°C is room temperature. If an object cools from $100$°C to $60$°C in $10$ minutes, what is the temperature at $t=20$ minutes?

A population $P$ satisfies $\frac{dP}{dt}=0.05P$. If $P(0)=200$, find $P(t)$.