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Differential Equations Unit 2: Modeling with first-order DEs

Differential equations model change in the real world. Set up the rate of change in words first, then translate.

Covers: Growth/decay, logistic, mixing problems, Newton's law of cooling

Study guide

Differential equations model real systems by describing how quantities change. Exponential growth and decay, dy/dt = ky, has solution y = y₀e^(kt): growth for k > 0 and decay for k < 0. For decay, the half-life is t½ = ln 2/|k|, and the same idea gives doubling time for growth. Radioactive decay, continuously compounded interest and simple population growth all fit this model.

Logistic growth, dP/dt = kP(1 − P/L), adds a carrying capacity L. Solutions approach L over time, with equilibria at 0 (unstable) and L (stable), and growth is fastest at P = L/2. The solution is P(t) = L/(1 + Ae^(−kt)).

Mixing problems track the amount A(t) of a substance in a tank: dA/dt = (rate in) − (rate out). The rate in is the incoming concentration times the inflow rate. The rate out is the tank's concentration, A/V, times the outflow rate. If the volume changes because inflow and outflow differ, V depends on t. These equations are usually linear and solved with an integrating factor.

Newton's law of cooling says an object's temperature changes at a rate proportional to the difference from its surroundings: dT/dt = −k(T − Tₛ). The solution is T(t) = Tₛ + (T₀ − Tₛ)e^(−kt), so the temperature approaches the surrounding temperature exponentially. Use two data points to find k.

With constant harvesting h, the logistic model becomes dP/dt = kP(1 − P/L) − h. Set the right side to 0 to find the new equilibria, which move inward from 0 and L. If h exceeds the maximum growth rate kL/4, no equilibrium remains and the population crashes. A quantity that is added at a constant rate r and lost at a rate proportional to the amount satisfies dx/dt = r − kx. Its solution approaches the steady state r/k exponentially. When the loss is radioactive decay, k = ln 2 divided by the half-life.

Key ideas

• Exponential growth or decay: rate proportional to amount.
• Newton's law of cooling: rate proportional to the temperature difference with the surroundings.
• Mixing problems: rate of change = rate in − rate out.
• Logistic growth adds a carrying capacity.
• Equilibrium solutions are constant solutions where y′ = 0.

Formulas and rules

Common mistakes

• Mixing problems: using the amount instead of the concentration for the outflow.
• Sign errors in Newton's law of cooling.

Words to know

Newton's law of cooling

dT/dt = −k(T − Tₛ)

Setup for a mixing problem

dA/dt = rate in − rate out

Half-life from decay constant k

t½ = ln 2 / k

Equilibrium solution

A constant solution where y′ = 0.

Outflow rate of salt in a mixing tank

(Concentration in tank) × (outflow rate) = (A/V)·r

Exponential growth/decay

dy/dt = ky → y = y₀e^(kt)

Doubling time

ln 2 / k for growth.

Logistic equation

dP/dt = kP(1 − P/L)

Carrying capacity

L, the stable equilibrium of the logistic model.

Newton's law of cooling solution

T(t) = Tₛ + (T₀ − Tₛ)e^(−kt)

Concentration in a tank

A(t)/V(t)

Stable vs. unstable equilibrium

Nearby solutions approach (stable) or move away (unstable).

Sample practice questions

19 more questions like these, from level 1 to 5, are in the practice set.