Roche

F(x, y, y0, y00) = 0
F y, y0, y00,
F=A(x)y00 +B(x)y0+C(x)y+g(x) = 0.
A(x)y00 +B(x)y0+C(x)y=f(x)∗
f(x), y
A(x)y00 +B(x)y0+C(x)y= 0 †
y, y0, y00.† ∗.
D[y(x)] = A(x)y00 +B(x)y0+C(x)y
D[y(x)] y(x)D[y(x)]
y(x)7→ D[y(x)]
λ µ y1y2x,
D[λy1(x) + µy2(x)] = λD[y1(x)] + µD[y2(x)].
D[λy1+µy2] = A(x)[λy100 +µy200] + B(x)[λy0
1+µy0 2] + C(x)[λy1+µy2].
λ µ
A(x)y00 +B(x)y0+C(x)y= 0 †
y1(x), y2(x), y(x) = λy1(x)+µy2(x)
λ µ y1(x), y2(x),
†, y(x) = C1y1(x)+C2y2(x)C1, C2
y1(x)
y2(x)y0
1y2−y1y0 2≡0.
y1(x)y2(x)y2=ky1y(x) =
C1y1(x) + C2ky1(x) = C0y1(x), C0
y0(x)
A(x)y00 +B(x)y0+C(x)y=f(x)
y0(x)
†.

y1(x)†,
y1(x)D[y1(x)] = 0. y(x) = y1(x)z(x).
z(x) : D[y(x)] = 0 z(x)≡1.
D[y(x)] = D[y1z] = A(x)[y1z00 + 2y0
1z0+y100z] + B(x)[y1z0+y0 1z] + C(x)y1z= 0
A(x)y1z00 + [2A(x)y0
1+B(x)y1]z0+zD[y1(x)] = 0.
D[y1(x)] = 0,
A(x)y1z00 + [2A(x)y1+B(x)y1]z0= 0
z≡1z0
z00
z0=−2A(x)y0
1(x) + B(x)y1(x)
A(x)y1(x)=−2y0
1
y1−B(x) A(x).
ln z0=−2 ln y1−ZB(x)
A(x)dx +K
z0=C2
y1(x)e−RB(x)
A(x)dx =C2ϕ(x)
z=C1Zϕ(x)dx +C1=C2ψ(x) + C1.
y=y1z,
y=C2y1(x)ψ(x) + C1y1(x).‡
† ‡
C1C2,‡ †
y1(x)y2(x)†
y2(x)
y1(x)†
y(x) = C1y1(x) + C2y2(x)
y1y2.
y1(x)y(x) = y1(x)z(x)z0
z0(x) = d
dx
y2(x)
y1(x).
z(x) = C2
y2(x)
y1(x)+C1.
‡y(x) = z(x)y1(x) = C2y2(x) + C1y1(x).
†

∗†
y(x) = u1(x)y1(x) + u2(x)y2(x)u1u2
D[y(x)] = f(x).
y0= (u0
1y1+u0 2y2)+(u1y0 1+u2y0 2). u0 1y1+u0 2y2=
0. y0=u1y0
1+u2y0 2. y00 =u0 1y0 1+u0 2y0 2+u1y00 1+u2y00 2.∗
D[y(x)] = A(x)[u0
1y0 1+u0 2y0 2] + u1D[y1(x)] + u2D[y2(x)]. D[y1(x)] = D[y2(x)] = 0,
A(x)[u0
1y0 1+u0 2y0 2] = f(x)
u1u2
(u0
1(x)y1(x) + u0 2(x)y2(x) = 0
u0
1(x)y0 1(x) + u0 2(x)y0 2(x) = f(x)
A(x).
u0
1(x)u0 2(x).
u0
1(x)u0 2)(x)y1(x)y0 2(x)−
y0
1(x)y2(x)y2(x)
y1(x)u0
i(x) = ϕi(x), i = 1,2,
ui(x) = Rϕi(x)dx+Kiy(x) = u1y1+u2y2
∗.
y1(x)†
∗
y00+y= 2 cos x. y00+y= 0
y1= cos x y2= sin x.
y=u1cos x+u2sin x
(u0
1(x) cos x+u0 2(x) sin x= 0
u0
1(x)(−sin x) + u0 2(x) sin x= 2 cos x
u0
1=−2 sin xcos x u0 2= 2 cos2x, u1=1
2cos 2x+C1u2=
x+1
2sin 2x+C2. y =xsin x+C0
1cos x+C0 2sin x
y(x)y00 +y= 2 cos x, y(a) = α
y0(a) = β.
y(a) = asin a+C0
1cos a+C0 2sin a=α, y0(a) = acos a+ sin a−C0 1sin a+
C0
2cos a=β. C0 1C0 2.
ay0+by =f(x)a6= 0.
ay0+by = 0. y =erx r=−b
a

ar +b= 0.
y=v(x)erx, v0=1
ae−rxf(x).
y(x) = erx Z1
ae−rxf(x)dx +C.
f(x)esx
y(x) = erx 1
aZe(s−r)xdx +C=1 a(s−r)esx +Cerx.
s=r, y(x) = x
aerx +Cerx.
f(x)n, P (x)
y(x) = Cerx +1
aQ(x)
Q(x)n,
Q(x) = erx Ze−rxP(x)dx.
Q n
aQ0+bQ =aP.
r= 0 b= 0
f(x) = esxP(x). P (x)y=esxz
ay00 +by0+cy = 0 a, b, c y =erx, r
y0=rerx y00 =r2erx y=erx (ar2+br+c)erx =
0.
erx ay00 +by0+cy = 0 r
ar2+br +c= 0
a, b, c
•r1r2.
y1=er1xy2=er2x. r2−r16= 0.
y=C1er1x+C2er2x
•r1=α+iβ, r2=α−iβ
er1x=e(α+iβ)x=eαx(cos βx +isin βx)
er2x=e(α−iβ)x=eαx(cos βx −isin βx).
Y1=1
2(y1+y2) = eαx cos βx Y2=1 2i(y1−y2) = eαx sin βx

y(x) = eαx[K1cos βx +K2sin βx]
K1, K2.
ϕ(K1, K2)A
Acos ϕ=K1Asin ϕ=−K2,
Acos(βx +ϕ) = Acos ϕcos βx −Asin ϕsin βx =K1cos βx +K2sin βx
y(x) = eαxAcos(βx +ϕ)
α=−b
2a,2aβ =√4ac −b2;A ϕ
y00 +y= 0
•ar2+br+c= 0 r=−b
2a. y =erxz,
az00 + (2ar +b)z0= 0, ϕ(r) = ar2+br +c
1
2ϕ00(r)z00 +ϕ0(r)z0= 0.
ϕ0(r) = 0 r z az00 = 0
z=K1x+K2
y(x) = erx[K1x+K2].
y00 + 2y0+y= 0. r2+ 2r+ 1 = (r+ 1)2= 0.
y=e−x(C1x+C2).
y00 −y0+y= 0. r2−r+ 1 = 0 1
2±i√3 2.
y=ex
2"K1cos x√3
2+K2sin x√3 2#.
y00 + 3y0+y= 0; r2+ 3r+ 1 = 0 −3±√5
2
y=e−3
2xK1ex√5 2+K2e−x√5 2.
Iθ00 +fθ0+cθ = 0,
I fθ0=fdθ
dt
dθ
dt ;cθ θ
θ= 0 I, f c
Ir2+fr +c= 0.