MATH 312 Section 2.4: Exact Differential Equations
Exact Differential Equations
Solving an Exact DE
Making a DE Exact
Conclusion
MATH 312 Section 2.4: Exact Differential Equations
Prof. Jonathan Duncan
Walla Walla University
Spring Quarter, 2008
Exact Differential Equations
Outline
Solving an Exact DE
Making a DE Exact
Conclusion
1 Exact Differential Equations 2 Solving an Exact DE 3 Making a DE Exact 4 Conclusion
Exact Differential Equations
Solving an Exact DE
Making a DE Exact
Conclusion
A Motivating Example
Our tools so far allow us to solve first-order differential equations which are separable and/or linear. Example Is the following differential equation separable or linear?
(tan x - sin x sin y )dx + (cos x cos y )dy = 0
After rewriting as shown, what do you notice?
dy sin x sin y - tan x
=
dx
cos x cos y
The equation is not separable.
The equation is not linear.
We need a new solution method for this DE!
Exact Differential Equations
Solving an Exact DE
Making a DE Exact
Conclusion
Working Backwards
We develop our method using Calculus notation.
Differentials Recall that if f (x, y ) has continuous first partials on some region of the xy -plane, then with z = f (x, y ) the differential is:
f
f
dz = dx + dy
x
y
Why is this of use? Recall our motivating example. Example Now, to solve
(tan x - sin x sin y )dx + (cos x cos y )dy = 0
we
find
an
f (x, y )
for
which
f x
=
(tan x
- sin x sin y )
and
f y
= (cos x cos y ), and
set
f (x, y ) = c
for
any constant
c
so
that
dz
= 0.
Exact Differential Equations
Solving an Exact DE
Making a DE Exact
Conclusion
Exact Differentials and Equations
We now formalize this type of solution with several definitions. Definition 2.3 A differential expression of the form M(x, y ) dx + N(x, y ) dy is an exact differential in a region R of the xy -plane if it corresponds to the differential of some function f (x, y ) defined on R.
Definition 2.3, Part II A first order differential equation of the form
M(x, y ) dx + N(x, y ) dy = 0
is an exact equation if the left side is an exact differential.
Solving an Exact Equation If the differential of f (x, y ) is M(x, y ) dx + N(x, y ) dy , then f (x, y ) = c is an implicit solution to the DE M(x, y ) dx + N(x, y ) dy = 0
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