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u
u(x) = f (x, u(x))
I,
u(x0) = u0.
f : R2 → R
x0 ∈ I u0
f
f (x, u) = g(x) h(u), g, h : R→ R
,
h
f
f (x, u) = a(x) u + b(x), a, b : R → R
.
a
b
x0
x0 u0
δ
a b
f
f
f (x, u) = a(x) u, b = 0.
g(x) = 1, h(u) = u,
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H (u(x)) = G(x) G(x) =
x
x0
a(y) dy, H (u) =
u
u0
dv
v .
a
x0
G
h
R
(−∞, 0)
(0,∞)
sgn u = sgn u0
H (u) = ln |u| − ln |u0| = ln | u
u0| = ln
u
u0.
ln u
u0=
x
x0
a(y) dy, u(x) = u0 exp(
x
x0
a(y) dy).
u
u0 = 0
I
a
u0 ∈ R x0
I
U (x, y) = exp( xy
a(z ) dz ), x, y ∈ I .
U
U (x, x) = 1, x ∈ I ,
U (x, y) U (y, z ) = U (x, z ), x,
y, z ∈ I ,
∂U (x, y)
∂x = a(x) U (x, y), x, y ∈ I ,
∂U (x, y)
∂y = −a(y) U (x, y), x, y ∈ I .
U
b = 0
u(x) = u0 U (x, x0), x ∈ I .
b = 0
a, b ∈ C (I ), I
, x0 ∈ I , u0 ∈ R.
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u(x) = a(x) u(x) + b(x), u(x0) = u0
u(x) = λ(x) U (x, x0),
λ
u
λ ∈ C 1(I )
U
u0 = u(x0) = λ(x0) U (x0, x0)
= λ(x0),
a(x) u(x) + b(x) = u(x) = λ(x) U (x, x0)
+ λ(x) ∂
∂xU (x, x0).
λ(x) U (x, x0) = b(x) λ(x) = b(x)
U (x0, x).
λ
λ(x) = b(x) U (x0, x), x ∈ I , λ(x0)
= u0.
U (x0, x) ∈ C (I )
I
λ(x) = u0 +
x
x0
b(z ) U (x0, z ) dz, x ∈ I .
λ
u
u(x) = u0 U (x, x0) +
x
x0
b(z ) U (x, z ) dz, x ∈ I .
u(x) = a(x) u(x), u(x0) = u0,
u(x) = a(x) u(x) + b(x), u(x0) = 0
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I
R
a, b ∈ C (I )
x0 ∈ I u0 ∈ R
u(x) = a(x) u(x) + b(x), u(x0) = u0
u
I
u(x) = u0 U (x, x0) +
x
x0
b(z ) U (x, z ) dz, x ∈ I ,
U : I × I →
R
U (x, y) = exp
x
y
a(z ) dz
, x, y ∈ I .
• x u + 5 u = 2 x, u(−1) = 1
• u = u2, u(0) = 1
• u = 1 + u2, u(0) = 0
• u = 3x2+4x+2
2u−2 , u(0) = 1
u(x) = a(x) u(x) + b(x) u(x)α
a, b ∈ C (I ), α ∈ R,
u
v(x) = u(x)1−α
