MAT 3237
Differential Equations
Section 2.4
Exact Equations
http://myhome.spu.edu/lauw
HW
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WebAssign 2.4
If encounter problems, type 𝑥 ∗ 𝑦4 for
𝑥𝑦4.
Actuary Presentation
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Recall
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Differentiation
Implicit Differentiation
Partial Differentiation
Recall
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Differentiation
𝑦 𝑥
𝑦 = sin 𝑥
𝑑𝑦

𝑑𝑥
= cos(𝑥)
Recall
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Implicit Differentiation
𝑦 𝑥
sin 𝑥𝑦 − 𝑦 = 0
d
d
sin
xy

y



0


dx
dx
 dy
 dy
cos  xy   x  y  

 dx
 dx
y cos  xy 
dy

dx 1  x cos  xy 
Recall
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Implicit Differentiation
𝑦 𝑥
Solution
sin 𝑥𝑦 − 𝑦 = 0
𝑑𝑦
,
𝑑𝑥
d
d
sin
xy

y



0


dx
dx
 dy
 dy
cos  xy   x  y  

 dx
 dx
Given
how to
recover sin 𝑥𝑦 = 𝑦?
y cos  xy 
dy

dx 1  x cos  xy 
D.E.
Recall
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Partial Differentiation
𝑓 𝑥, 𝑦
𝑓 𝑥, 𝑦 = sin 𝑥𝑦
𝜕𝑓

𝜕𝑥
= 𝑦 cos(𝑥𝑦)
Preview
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Solving exact D.E., which always* give
implicit solutions.
The solution process involves the
antiderivatives of partial derivatives.
View the process as the reverse
operation of implicit differentiation
Exact D.E.
M ( x, y )dx  N ( x, y)dy  0
is exact if there is a function 𝑓 such that
f
f
 M and
N
x
y
Exact D.E.
M ( x, y )dx  N ( x, y)dy  0
is exact if there is a function 𝑓 such that
f
f
 M and
N
x
y
The general solutions* is
f ( x, y )  C
See Example
Below
Criterion
f
f
 M and
N
x
y
Use the definition to check the exactness
of a D.E. is difficult. Instead, we use
Theorem 2.1
M ( x, y )dx  N ( x, y)dy  0
is exact if and only if
M N

Make sense? y x
Criterion
M N

Make sense? y x
f
f
 M and
N
x
y
Example 1
dy
sin y  ( x cos y  2 y )  0
dx
(a)
(b)
(c)
Verify that the D.E. is exact
Solve the exact D.E.
Use implicit differentiation to verify the
solutions
Example 1(a)
dy
sin y  ( x cos y  2 y )  0
dx
Example 1(b)
dy
sin y  ( x cos y  2 y )  0
dx
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To solve the D.E., we need to find the
function 𝑓(𝑥, 𝑦)
The process requires the antiderivative
of partial derivatives
You will use this same technique in
multivariable calculus
Antiderivatives:
Derivatives Vs Partial Derivatives
y  x2 1
dy
 2x
dx
y  x 2  251
dy
 2x
dx
2
2
xdx

x
C

Antiderivatives:
Derivatives Vs Partial Derivatives
y  x2 1
z  x2 y  y
dy
 2x
dx
y  x 2  251
z
 2 xy
x
z  x 2 y  y 3  e5 y cos y  10
dy
 2x
dx
2
2
xdx

x
C

z
 2 xy
x
2
2
xydx

x
y  ????

Antiderivatives:
Derivatives Vs Partial Derivatives
y  x2 1
z  x2 y  y
dy
 2x
dx
y  x 2  251
z
 2 xy
x
z  x 2 y  y 3  e5 y cos y  10
dy
 2x
dx
2
2
xdx

x
C

z
 2 xy
x
2
2
xydx

x
y  ????

No new
notations…
Example 1(b)
dy
sin y  ( x cos y  2 y )  0
dx
f
f
 M and
N
x
y
Example 1(c)
x sin y  y 2  C
d
d
 x sin y  y 2   C 
dx
dx
Pay Attention to the
Presentation
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The connection statements are curial for
your audience to understand your
solutions.
You do not need to use the exact
wordings, but you need put down the
arguments carefully.
Expectations
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The connection statements are curial for
your audience to understand your
solutions.
You do not need to use the exact
wordings, but you need put down the
arguments carefully.
Common misconceptions
𝑓 is not the general solutions, rather,
𝑓(𝑥, 𝑦) = 𝐶 is the general solutions
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In example 1, 𝑥siny + 𝑦 2 is not the
general solutions
• Not a relation between 𝑥, 𝑦