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UNIT 7 · 6–9%

Differential Equations

equations where the unknown is a function 🌊

🧭 what is a DE?

A differential equation relates a function to its derivatives. Solutions can be explicit y = … or implicitly defined curves / families (+C).

🏔 slope fields

For dy/dx = f(x,y): at each grid point, draw a tiny segment with slope f(x,y). Solution curves trace the tangent directions. Helps visualize families of solutions and particular solutions through initial points.

🔁 separation of variables

Rewrite dy/dx = g(x)h(y) as (1/h(y)) dy = g(x) dx, integrate both sides, isolate y if possible, use initial condition to find C.

▶ Euler's method BC

Starting at (x₀,y₀) with dy/dx = f(x,y):   yn+1 = yn + h · f(xn, yn),   xn+1 = xn + h

Step size h smaller → usually better accuracy but more arithmetic.

📈 logistic growth BC

dP/dt = k P (M − P) → limiting population M, S-shaped curve

Carrying capacity M; growth slows as P approaches M. Know the shape: increasing, concave up then down, inflection where P = M/2 for standard form.

✏️ quick practice

  1. Sketch slopes for dy/dx = x + y at a few points near the origin.
  2. Solve dy/dx = ky with y(0) = y₀ (exponential model).
  3. One Euler step for dy/dx = x − y starting at (0, 1) with h = 0.5.
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applications of integration = geometry with integrals 📦