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y ′ = 2xy − x y(0) = 3/2
y(0) = 3/2
ω(x, y) = p(x, y) dx + q(x, y) dy = (−2xy + x) dx + dy = 0.
∂yp − ∂xq = −2x = −2x q
h(x) = e −2x dx = e−x2 . hω R2
V
(x0, y0) ∈ R2 γ(t)
x0
V (x0, y0) = hω = e −x/2 y0
x dx +
γ
V
0
0
e −x/2 dy = − 1
2 e−x2
+ ye −x2
= (y − 1/2)e −x2
.
h(x)ω(x, y) = e −x2
(−2xy + x) dx + e −x2
dy = d(ye −x2
) + e −x2
x dx
= d(ye −x2
) + d − 1
2 e−x2
= d (y − 1/2)e −x2
.
(y − 1/2)e−x2 = c c ∈ R
y(x) = cex2 + 1/2 y(0) = 3/2 c = 1 y(x) = ex2 + 1/2
R +∞ x → ±∞
x > 0
x < 0 x = 0 3/2
dy
dx = − 1 − x2y x2 .
y − x3
lim |y(x)| = +∞ 0
x→0
ω(x, y) = (1 − x 2 y) dx + (x 2 y − x 3 ) dy = 0.
∂yp(x, y) − ∂xq(x, y) = 2x 2 − 2xy = − 2
q(x, y),
x
λ(x, y) = e − 2
x dx = 1
.
x2