C.3.2 - Separable Equations
Calculus - Santowski
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Lesson Objectives


1. Solve separable differential equations
with and without initial conditions
2. Solve problems involving exponential
decay in a variety of application
(Radioactivity, Air resistance is
proportional to velocity, Continuously
compounding interest, Population growth)
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(A) Separable Equations

So far, we have seen differential
equations that can be solved by
integration since our functions were
relatively easy functions in one variable

Ex. dy/dx = sinx - 1/x
Ex. dv/dt = -9.8

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(A) Separable Equations

In Ex 1, we simply
evaluated the indefinite
integral of both sides
dy
1
 sin x 
dx
x
dy 

1 
 dx dx   sin x  x dx
y  cos x  ln x  C


But what about the
equation dy/dx = -x/y?
If we tried finding
antiderivatives or
indefinite integrals ….
dy
x

dx
y
dy 
 dx dx 
 x 
  y dx
y  ???

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(A) Separable Equations
If we have equations in the form of
1
dy
g(y) 
where
, then
 f (x)  g(y)
h(y)
dx
dy f (x)

such that h(y)  0

dx

h(y)
Then we can rewrite the equation as

h(y)dy  f (x)dx
 and integrate as before
 h(y)dy   f (x)dx
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(B) Examples



dy
x
Given the DE
  and y(2)  5
dx
y
(a) Solve
(b) Graph

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(B) Examples



2

y

x
y and y
Given the DE
 3 4e
3
(a) Solve

(b) Graph
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(B) Examples

Given the DE
dy
dy
yy
 xy 
and y(2) 1
dx
dx



(a) Solve
(b) Graph
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(B) Examples

Given the DE
dy
 xeyx and y(0)  2
dx
(a) Solve
 (b) Graph

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(B) Examples
2



Given the DE
dy
6x

dx 2y  cos y
(a) Solve given y 1  
(b) Graph
 the solutions on a slope field
diagram

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(B) Example

Here is the graphic
solution for
dy
6x 2

dx 2y  cos y

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(C) - Application - Exponential
Growth

Write a DE for the statement: the rate of
growth of a population is directly
proportional to the population

Solve this DE
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(C) - Application - Exponential
Growth

Write a DE for the
statement: the rate of
growth of a population
is directly proportional
to the population
dP
 P or
dt

Solve this DE:
dP
 kP
dt
dP
 kP
dt
dP
 P   kdt
ln P  kt  C
P(t)  e kt C  eC e kt
P(t)  Ce kt

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(D) Examples

The population of bacteria grown in a culture
follows the Law of Natural Growth with a
growth rate of 15% per hour. There are
10,000 bacteria after the first hour.

(a) Write an equation for P(t)
(b) How many bacteria will there be in 4
hours?
(c) when will the number of bacteria be
250,000?


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(D) Examples

The concentration of phosphate pollutants in a lake
follows the Law of Natural Growth with a decay rate
of 5.75% per year. The phosphate pollutant
concentrations are 125 ppm in the second year.

(a) Write an equation for P(t)
(b) What will there be phosphate pollutant
concentration in 10 years?
(c) A given species of fish can be re-introduced into
the lake when the phosphate concentration falls
below 35 ppm. When can the fish be re-introduced?


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(E) Homework



See handouts
(1) Salas, §9.2, p457, Q1-20
(2) Dunkley, §7.5, p371, Q1-7
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