Ch 12/Applications of Differential Equations/H.K. Dass 3.14-3.17

Applications of Differential Equations

DEs model real-world phenomena: electrical circuits, Newton's cooling, orthogonal trajectories, population growth/decay, spring-mass systems, and more.

IElectrical Circuits

L-R Series Circuit: . Solution: .


R-C Series Circuit: .

Circuit Examplemedium

R = 15 ohms, L = 10 H, E = 90 V. Find current after 2 seconds.

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DE
. Solution: .
At t = 2
A.

IINewton's Law of Cooling

. Solution: .

Cooling Examplemedium

Body cools from 100C to 75C in 1 min (air at 25C). Find T at 3 min.

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Setup
. At : .
At t = 3
C.

IIIOrthogonal Trajectories

Two families are orthogonal trajectories if they cut at right angles.


Rule: Find DE of family, replace by , solve.

OT of xy = ceasy

Find the orthogonal trajectories of .

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DE of family
. Replace: .
Solve
(hyperbolas).
OT of y = ax^2medium

Find OT of the parabolas .

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DE
. OT: .
Solve

IVGrowth, Decay, and Other Applications

. Solution: . Growth if , decay if .

Populationeasy

Population doubles in 50 years. When does it triple?

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Find k
.
Tripling time
years.
Decayeasy

Radium: 5% gone in 50 years. How much after 100 years?

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At 50 years
.
At 100 years
. Answer: 90.25% remains.
Spring-massmedium

Solve with .

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A.E.
. .
Apply conditions
.
Answer
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