Rucete ✏ AP Calculus In a Nutshell
9. Differential Equations
This chapter focuses on writing, analyzing, and solving differential equations using graphical, numerical, and analytical methods. BC topics include Euler’s Method and logistic growth models.
A. Basic Definitions
• A differential equation (d.e.) contains one or more derivatives
• A general solution includes a constant C and represents a family of curves
• A particular solution satisfies both the d.e. and an initial condition
• The domain of a solution must be considered — some equations have vertical tangents or undefined behavior at specific points
Examples of Rate Models:
– Population growth: P′ = 0.0325P – Radioactive decay: Q′ = −0.000275Q – Stimulus-response: dy/dx = 1/x – Gravitational motion: a = −g
B. Slope Fields
• A slope field shows tiny line segments whose slopes are defined by the d.e. • Slope fields help visualize solution curves
Example: y′ = y – Slope at any point (x, y) is y – General solution: y = Ceˣ – Steeper slopes for larger y-values
Example: y′ = cos(x) – General solution: y = sin(x) + C
Example: y′ = x – General solution: y = (1/2)x² + C
Example: x² + y² = r² → dy/dx = −x/y – Represents circles; solution domain is limited where y ≠ 0
C. Euler’s Method (BC)
• Numerical method to approximate solutions of d.e.’s step-by-step
• Start with a point (x₀, y₀), then use the slope to estimate next y-value:
yₙ₊₁ = yₙ + f(xₙ, yₙ)·Δx
Key Observations:
– Smaller Δx → more accurate – May fail near vertical asymptotes or discontinuities – Compare to true solution if available
Example: y′ = 3/x, y(1) = 0, Δx = 0.5 → Approximate y-values up to x = 3 → Compare to y = 3 ln(x)
D. Separable Differential Equations
• Can be written as: dy/dx = g(x) · h(y)
• Separate and integrate both sides: ∫1/h(y) dy = ∫g(x) dx
• Add +C and solve for y (if possible)
Example: dy/dx = y/x → dy/y = dx/x → ln|y| = ln|x| + C → y = Cx
E. Exponential Growth and Decay
• General form: dy/dt = ky → y = y₀e^(kt)
• Use initial conditions to solve for k
Example: dy/dt = −0.275y, y(0) = 100 → y = 100e^(−0.275t)
F. Newton’s Law of Cooling
• Temperature T(t) changes at a rate proportional to difference from surrounding temp:
dT/dt = −k(T − Tₐ)
• Solution: T(t) = Tₐ + (T₀ − Tₐ)e^(−kt)
G. Logistic Growth (BC)
• For populations with carrying capacity M:
dP/dt = kP(M − P) or dP/dt = kP(1 − P/M)
• General solution: P(t) = M / [1 + Ae^(−kMt)]
• Properties:
– Fastest growth occurs at P = M/2 – Growth slows as P approaches M – Horizontal asymptote at y = M
Example: P(0) = 100, carrying capacity M = 500 → Solve for A, k if data is given
In a Nutshell
Differential equations describe dynamic change. Graphical (slope fields), numerical (Euler’s Method), and analytical (separation, exponential/logistic) techniques provide solutions. BC topics like logistic growth and Euler's approximation extend the power of calculus into modeling realistic, bounded systems.