Unit 7: Differential Equations
Differential equations describe how a quantity changes rather than giving the quantity directly. In AP Calculus, the focus is on slope fields, separable equations, logistic models, and interpreting solution behavior.
General and particular solutions
A differential equation relates a function and its derivatives.
Example:
\[\frac{dy}{dx} = 3x^2\]has general solution
\[y = x^3 + C.\]If an initial condition is given, such as \(y(1)=5\), you solve for the constant to get the particular solution.
Slope fields
A slope field shows small line segments representing \(dy/dx\) at many points.
To analyze a slope field:
- look for where slopes are zero,
- look for where slopes are positive/negative,
- identify equilibrium solutions,
- sketch a solution curve that follows the segment directions.
[Image Placeholder: slope field with equilibrium solution and sample integral curves]
Euler’s method
Starting from \((x_0,y_0)\) with step size \(h\):
\[y_{n+1} = y_n + h f(x_n,y_n)\]where
\[\frac{dy}{dx} = f(x,y).\]Separable differential equations
If
\[\frac{dy}{dx} = g(x)h(y),\]rewrite as
\[\frac{1}{h(y)}\,dy = g(x)\,dx\]and integrate both sides.
Exponential growth and decay
If the rate of change is proportional to the amount present:
\[\frac{dy}{dt} = ky\]then
\[y = Ce^{kt}.\]Logistic differential equation
The logistic model is
\[\frac{dy}{dt} = ky\left(1-\frac{y}{L}\right)\]where \(L\) is the carrying capacity.
Behavior:
- equilibrium solutions at \(y=0\) and \(y=L\),
- growth is fastest near \(y=L/2\),
- solutions below \(L\) increase toward \(L\).
Equilibrium solutions and stability
Equilibrium solutions are constant solutions where \(dy/dx = 0\).
Stability:
- stable if nearby solutions move toward it,
- unstable if nearby solutions move away.
For autonomous equations \(dy/dx = f(y)\), a sign chart on \(f(y)\) is an efficient way to classify equilibria.
Second derivative from a differential equation
If
\[\frac{dy}{dx} = f(x,y),\]then
\[\frac{d^2y}{dx^2}\]often comes from differentiating implicitly:
\[\frac{d^2y}{dx^2} = \frac{d}{dx}[f(x,y)].\]This helps determine concavity of solution curves.
Common mistakes
- Separating variables incorrectly.
- Forgetting the constant of integration.
- Solving for the constant before using the initial condition carefully.
- Sketching slope-field solutions that cross each other or violate the displayed slope directions.