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y ′ = f(x, y)
A
A
F (x, y, y ′ ) = 0
y ′ = f(x, y)
˙y = p(t)q(y)
p : I → R q : J → R I, J R q(y0) = 0
y(t) = y0 q(y)
q(y(t)) = 0 η = y(t)
y(t0) = y0
y dη
y0 q(η) =
t
p(τ)dτ.
t0
y ′ = f(ax + by) f a, b ∈ R \ {0} z = ax + by
z ′ = a + bf(z) z(x)
y(x) = z(x)−ax
b
y ′
y
= f
x
0 z =
y
x z′ = f(z)−z
x
y(x) = xz(x)
M(x, y) dx + N(x, y) dy = 0
M, N y = xz
x = 0 x = yz y = 0
y ′
= f
f a, b, c, a ′ , b ′ , c ′
ax+by+c
a ′ x+b ′ y+c ′
ab ′ − a ′ b = 0 c, c ′ ax + by + c = 0