Modeling with First Order DEs
Translating real-world problems into differential equations. Rate In minus Rate Out.
Introduction
The power of differential equations comes from modeling rates of change. The universal conservation law is our starting point:
Whether it's salt in a tank or money in a bank, this principle holds.
Mixing Problems
Scenario
A tank holds gallons of water with lbs of salt. Brine flows in at rate with concentration , and the mixture flows out at .
The Setup
- Rate In: (lbs/min).
- Rate Out: (lbs/min).
- Volume: .
Visual: Concentration Equilibrium
Interactive: Tank flushing
Worked Examples
Example 1: Mixing Problem
Tank has 500 gals pure water. Brine (2 lb/gal) enters at 5 gal/min. Mix leaves at 5 gal/min. Find Salt at 10 mins.
1. Setup:
Rate In: lb/min.
Rate Out: lb/min (Volume is constant).
DE: .
2. Solve (Linear):
.
IF: .
.
.
.
3. Initial Condition:
.
.
4. At t=10:
lbs.
Example 2: Population Growth
A population P grows at rate proportional to P. Initial pop 100, doubles in 5 hours.
1. Model:
.
2. Find parameters:
.
.
.
Example 3: Falling Body (Air Resistance)
Mass kg falls with drag . Initial .
1. Newton's 2nd Law:
(Down is positive).
.
.
2. Solve (Linear):
.
.
3. Terminal Velocity:
As , m/s.
Practice Quiz
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