AP Calculus AB/BC - Unit 7: Differential Equations

Solving Differential Equations:

When you see the following phrase:

Use the general solution:

$$y = Ce^{kt}$$

Steps for Solving a Differential Equation:

1. Separate variables first.
2. Integrate.
3. Add $+C$ to one side.
4. Use initial conditions to find $C$.
5. Write the equation in the form of $y = f(x)$.

Logistic Growth Model (BC Only):

The differential equation for logistic growth is:

$$\frac{dy}{dx} = k y \left( 1 - \frac{y}{L} \right)$$

The general solution to this equation is:

$$y = \frac{L}{1 + be^{-kx}}$$

where $L$ is the carrying capacity, $k$ is the growth rate, and $b$ is a constant determined by initial conditions.