Calc2: 7.1 Modeling with Differential Equations

What is a differential equation?

A differential equation is an equation that contains an unknown function and one or more of its derivatives:contentReference[oaicite:0]{index=0}.

What is meant by “modeling with a differential equation”?

It means describing how a quantity changes over time or space by relating its rate of change (derivative) to the quantity itself through an equation:contentReference[oaicite:1]{index=1}.

State the population growth model.

The population growth model assumes the rate of change of population is proportional to the population itself:

[

\frac{dP}{dt} = kP

]

where (k>0) is the proportionality constant:contentReference[oaicite:2]{index=2}.

What is the general solution to (\frac{dP}{dt}=kP)?

[

P(t) = Ce^{kt}

]

where (C) is the initial population (P(0)):contentReference[oaicite:3]{index=3}.

What kind of growth does (\frac{dP}{dt}=kP) model?

Exponential growth — the population increases at a rate proportional to its current size, assuming unlimited resources:contentReference[oaicite:4]{index=4}.

What is the logistic differential equation?

To account for limited resources, the logistic model assumes

[

\frac{dP}{dt} = kP\left(1 - \frac{P}{M}\right)

]

where (M) is the carrying capacity:contentReference[oaicite:5]{index=5}.

Explain the behavior of the logistic model when (P < M) and (P > M).

If (P

What are equilibrium solutions?

Constant solutions where (\frac{dP}{dt}=0). For the logistic equation, these are (P=0) and (P=M):contentReference[oaicite:7]{index=7}.

What is the carrying capacity (M)?

The maximum population that an environment can sustain indefinitely; beyond (M), resources are insufficient and population decreases:contentReference[oaicite:8]{index=8}.

Describe the typical shape of the logistic growth curve.

It is sigmoidal (S-shaped): initially exponential, then slowing as it approaches the carrying capacity asymptotically:contentReference[oaicite:9]{index=9}.

What is an initial-value problem (IVP)?

A problem in which a differential equation is given along with an initial condition, such as (P(0)=P_0), to find a specific solution:contentReference[oaicite:10]{index=10}.

Give an example of an IVP using population growth.

[

\frac{dP}{dt}=kP, \quad P(0)=P_0

]

whose solution is (P(t)=P_0e^{kt}):contentReference[oaicite:11]{index=11}.

What is the meaning of (k) in population models?

The constant (k) is the proportional growth rate — if (k>0), the population grows; if (k<0), it decays.

What is the general definition of the order of a differential equation?

The order is the highest derivative that appears in the equation (e.g., (\frac{d^2x}{dt^2}) means second-order):contentReference[oaicite:12]{index=12}.

What is the second-order differential equation for a spring-mass system?

From Hooke’s Law and Newton’s Second Law:

[

m\frac{d^2x}{dt^2} = -kx

]

where (k) is the spring constant and (m) is the mass:contentReference[oaicite:13]{index=13}.

What are typical solutions to (m\frac{d^2x}{dt^2}=-kx)?

Trigonometric functions — the general solution is (x(t)=A\cos(\omega t)+B\sin(\omega t)), representing oscillatory motion:contentReference[oaicite:14]{index=14}.

Explain how mathematical modeling is used in this context.

Modeling involves identifying quantities that change, hypothesizing relationships among rates and values, forming differential equations, and analyzing or solving them for predictions:contentReference[oaicite:15]{index=15}.

Give an example of another real-world model that uses differential equations.

Newton’s Law of Cooling:

[

\frac{dT}{dt} = -k(T - T_s)

]

where (T_s) is surrounding temperature:contentReference[oaicite:16]{index=16}.

What is the learning-rate model from psychology?

[

\frac{dP}{dt} = k(M - P)

]

where (P(t)) is performance at time (t), (M) is the maximum performance, and (k>0) is a constant:contentReference[oaicite:17]{index=17}.