Math 334 Lecture #8
§
2.6: Exact First Order ODE’s
The first order ODE
2
x
+
y
2
+ 2
xy
dy
dx
= 0
is not linear, nor is it separable.
Notice, however, that since the partial derivatives of
ψ
(
x, y
) =
x
2
+
xy
2
are
∂ψ
∂x
= 2
x
+
y
2
and
∂ψ
∂y
= 2
xy,
the ODE can be written in the form
∂ψ
∂x
+
∂ψ
∂y
dy
dx
= 0
.
By the Chain Rule, the ODE simplifies to
d
dx
ψ
(
x, y
)
= 0
,
integration of which gives
ψ
(
x, y
) =
C
as a one parameter family of (implicitly defined) solutions of the ODE.
Geometrically, the level curves of
ψ
are integral curves of the ODE.
[There is no guarantee this is one parameter family of solutions is a general solution.]
A first order ODE of the form
M
(
x, y
) +
N
(
x, y
)
dy
dx
= 0
or
M
(
x, y
)
dx
+
N
(
x, y
)
dy
= 0
is called
exact
if there is a continuously differentiable function
ψ
(
x, y
) such that
M
(
x, y
) =
∂ψ
∂x
and
N
(
x, y
) =
∂ψ
∂y
.
Solutions of an exact first order ODE are given by
ψ
(
x, y
) =
C
.
The presence of the arbitrary constant
C
means that IVP’s may be solved.
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 Spring '11
 Smith
 Derivative, three weeks, ydx

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