A differential equation relates a function to its rate of change
A differential equation like dy/dt = ky relates a quantity to its own derivative — it describes how something changes rather than the quantity directly. You model a scenario by translating a described rate into an equation, and you verify a candidate solution by differentiating it and substituting back to confirm both sides agree. A general solution carries an arbitrary constant; a particular solution uses an initial condition to determine it.
ModelingVerifyingGeneral vs. Particular
Key Concept 2
Slope fields visualize solutions; separation of variables solves them
A slope field plots short segments whose slopes equal dy/dx at each point, letting you see the shape of solution curves and reason about their behavior without solving. To solve analytically, separation of variables rewrites dy/dx = g(x)h(y) as (1/h) dy = g dx and integrates both sides; applying the initial condition then gives the particular solution. Solution curves always follow the slope field’s flow.
Slope FieldsSeparationInitial Condition
Key Concept 3
Exponential and logistic models describe real growth
When a rate is proportional to the amount, dy/dt = ky, and the solution is exponential: y = y₀eᵏᵗ (growth if k > 0, decay if k < 0). BC students also use Euler’s method to approximate solutions numerically with tangent-line steps, and the logistic model dP/dt = kP(1 − P/L) for bounded growth that slows as P nears the carrying capacity L, growing fastest at P = L/2.
ExponentialEuler (BC)Logistic (BC)
Differential equation
An equation relating a function to its derivatives.
Basics
Solution
A function that satisfies the differential equation.
Basics
Verifying a solution
Substituting a candidate (and its derivatives) to check equality.
Basics
General solution
The family of solutions containing an arbitrary constant C.
Solutions
Particular solution
A specific solution found using an initial condition.
Solutions
Initial condition
A point (x₀, y₀) the solution must pass through.
Solutions
Slope field
Segments of slope dy/dx drawn across the plane.
Slope Fields
Solution curve
A curve tangent to the slope field through a given point.
Slope Fields
Separation of variables
Rewriting (1/h) dy = g dx and integrating each side.
Solving
Constant of integration
The +C added when integrating; found via the initial condition.
Solving
Exponential model
dy/dt = ky with solution y = y₀eᵏᵗ.
Models
Growth / decay constant
k in dy/dt = ky; positive grows, negative decays.
Models
Euler’s method (BC)
yₙ₊₁ = yₙ + f(xₙ, yₙ)Δx; a numerical approximation.