Lecture Notes in Differential Equations - Bruce E. Shapiro

21
where M(t) = −a(t) and N(y) = 1/b(y) are new names that we give our
functions. This gives us the standard form for a separable equation
M(t)dt + N(y)dy = 0 (3.36)
The reason for calling this the standard format will become more clear when
we study exact equations. To continue deriving our test for separability we
rearrange 3.36 as
dy
dt = −M(t)
(3.37)
N(y)
Recalling the standard form of an ordinary differential equation
we have
Since M is only a function of t,
∂M
∂t
dy
dt
= f(t, y) (3.38)
f(t, y) = − M(t)
N(y)
and because N is only a function of y,
Similarly
and
The cross-derivative is
Hence
(3.39)
= M ′ (t) = dM dt , ∂M
∂y = 0 (3.40)
∂N
∂t = 0,
∂N
∂y = N ′ (y) = dN
dy
f t = ∂f
∂t = (t)
−M′ N(y)
f y = ∂f
′ ∂y = −M(t)N (y)
N 2 (y)
f ty = ∂2 f
∂t∂y = (t)N ′ (y)
−M′ N 2 (y)
(
ff ty = − M(t) ) (
− M ′ (t)N ′ )
(y)
N(y) N 2 (y)
(
= − M ′ ) (
(t)
− M(t)N ′ )
(y)
N(y) N 2 (y)
(3.41)
(3.42)
(3.43)
(3.44)
(3.45)
(3.46)
= f t f y (3.47)