MATH 312 Section 2.4: Exact Differential Equations - Walla Walla University

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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