Matakuliah
Tahun
: D0024/Matematika Industri II
: 2008
Matematika
Pertemuan 13
Persamaan Diferensial Eksak
Consider a first-order ODE in the slightly different form
(
1
)
Such an equation is said to be exact if
(
2
)
This statement is equivalent to the requirement that a conservative field
exists, so that a scalar potential can be defined. For an exact equation, the
solution is
(
3
)
where
Bina Nusantara
is a constant.
A first-order ODE (◇) is said to be inexact if
(
4
)
For a nonexact equation, the solution may be obtained by defining an
integrating
factor
of (◇) so that the new equation
(
5
)
(
6
)
or, written out explicitly,
Bina Nusantara
satisfies
This transforms the nonexact equation into an exact one. Solving the last
equation for
gives
Therefore, if a function
found, then writing
satisfying equation can be
(
9
)
(
1
0
)
in equation (◇) then gives
(
1
1
)
which is then an exact ODE. Special cases in which
can be found include
-dependent,
-dependent, and -dependent integrating factors.
Bina Nusantara
(
8
)
• Contoh-contoh
• Kerjakan latihan dalam modul soal
Bina Nusantara