Exact Equations, p' 6772 2'4

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Exact Equations, p. 67-72 (2.4)

• OBJECTIVES
• Define linear DE
• Define exact DE
• Solve an exact DE

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• is a 1st order DE.
• is a 1st order DE.
• linear a DE when f is a linear function
  of , p. 4.
• A linear DE may be written in the form
• nonlinear a DE that is not linear

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• Consider the separable DE y dx x dy 0
• y dx x dy
• ln x ln y k , k constant
• ln x ln y k

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• Note the solution of y dx x dy 0 is xy c.
• Let f (x, y) xy. Then and
• This is called an exact equation.
• Substitution on the left hand side yields
• Recall the definition of the Total Differential
  from Calculus III (Calculus Text, p. 916),
• If z f (x, y) , then the total differential of
  the dependent variable z is

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• Consider the exact DE y dx x dy 0.
• We may assume , and f (x, y) C,
• C constant
• because and
• Integrating f (x, y) xy g(y),
• Differentiating w.r.t. y
• But we know
• Solution xy k C or xy c, c C k

g(y) arbitrary function
g(y) k constant
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Definition 2.3, p. 68 exact differential a
differential expression M(x, y) dx N(x, y)
dy in region R of the xy-plane corresponding to
a differential of function f (x, y). exact
equation a first order differential equation of
the form M(x, y) dx N (x, y) dy 0 where the
RHS is a differential expression.
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Theorem 2.1 Criterion for a Exact Differential,
p. 68 Let M(x, y) , N(x, y), Mx, My, Nx, Ny, be
continuous in region R defined by a lt x lt b ,
c lt y lt d. Then a necessary and sufficient
condition that M(x, y) dx N (x, y) dy
be an exact differential is

• Proof i) (exactness equality)
• If M (x, y) dx N (x, y) dy is exact, then
• Then and
• by continuity.

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• Proof ii) (exactness equality)
• If , then function
• Integrating w.r.t. x, .
• Differentiating w.r.t. y,
• Rearranging
• Integrating w.r.t. y,
• Substitution

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Method of solution, p. 68

• a) If M dx N dy 0 is exact , then
• Integrating w.r.t. x ()
• b) Use () to compute and substitute N to
  solve for
• c) Integrate and substitute into ().
• d) f (x, y) C is the solution to M dx N dy
  0.

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Homework

• p. 10 19-20
• p. 17 11-14
• p. 28-29 7-8
• p. 73-74 1-39 alternate odd
• Read p. 75-78 (2.5)
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