ページ3:

2nd order ODE,
y" + pixsy" + pixsy = rix)
=>
存在 2 个独立解 yy
,
→通解:
Y₁ = cy₁ + cy₂
二阶常 係数齐次 linear ODE:
y" + ay' + by =
= 0
n
相异实根 ai-4b> 0 =>
y₁
+
厶三种根
重根 a²-46=0
-%
=> y₁ = (c₁ + c₂x) e
-x
③相异複数根 ai-46-0
√a' - 4b
=>
= e (A cos wx + B sin wx)
W =
pt
Case distinct real root
利用
阶齐次 ODE (常数係数)
y + y = f f dy = f h dx x x x Inty = -hate the great.
EX
2x
= e
为方程式解
y" = x*e**
①令
y =
y' = λ ex
© π y " + ay' + by
2%
= 0
xe axe be^x = 0 A = x² + a λ' + b = o
+
-a±
7-9√9-46
2
(特徵方程式)
入=
-a+√a-46
2
y₁ = e₁₁x
入_ =
2
y₁ = e₁₁x
·Af: y = c.y. + C₂ Yo
=
c, ex
2,%
Case 2
real double root
-a±√9-46
λ =
λ = -A
-x
利用降阶法获得ă
ky₁ = uy₁
ý uy, tuy
代回
非常教
u"y₁ + "' (zy,' + ay₁)
=0
1 + 4 (y," + ay " + by₁ ) = 0
Case 3
complex
root
.. y " + ay + by = 0
+
+
(u'y₁ - zuy, + uy₁") + a (uy₁ - uy;) + buy.
= 0
Y₁ = e
- 1x (A cos wx
-a±√9-46
λ =
'
2
+ B sin wx) W =
其中a²-46< 0
4b-a
2
=>
u" = o u' = c — u = cx+c.
7/4 c
.. u = x
'
y₁ = xy,

ページ4:

λ₁ = = 2 + wi 入望
wi
=
-a-√46-a √- - -√4-=-=-14-
2
=
+
√46-a³
wi, w
+wi)x
y₁ = e²².
'
y
((1+wi)x
Y₁ = c₁ y₁ + C₂ y₁ =
= e
-1%
^ [C, (cos wx + i sin wx) + C₂ (cos wx + isin wx
c, e(+wi)x
+ C₂ e (+wi)x
- AX
= e
| wx)]
= e
- ** [ (C‚+ C.) co:
(c. ex + C₂ etx)
C,+ C₁) cos wx + (C. i-C₁i) sin wx
尤拉公式:
e = cose + isine
wx]
(ex y" + 3y²+ y
zy. e^x
入+ 37 + 1 = 0
(ex y" + y²+ y = o
-y' = ^e^^y" = x^e^x
1+3Ace-
-3+√5
y₁ =
-3-15
y-
X*+ 2 + 1 = (特徵方程)
λ=-1211-4
-147
--319-4-3115
↳ RLC circuit
:
V₁ = IR
V₁ = I'L V₁ =
Q
=
V₁ + V₁ + V₁ = E
+
E = o
IR I'L + Ide
=
0
t
二阶常数係数非齐次linear
ODE:
ay' + by
#32 +9
= (x)
①求齐次解
-
Y₁ = ex (A cos wx + B sin wx), w
I de
· I'R + I'L +
I
=
0
令r(x) = 0
'
"y" + ay' + by =
= 0
67 Af
ex
@求非齐次解
基本法则
待定係数法
修正法则
y"
+ ay' + by
总和法则
= 1(x)
+
L
I" - I'R
I
+
y" +by+9y6x"
(ex
E-0
R-100 LHC SF
[15]
10151
25-010
LC
= 0
x²+6+9+0
71+3)-0
11-163)
x--3k
(181)
basic rule
- KKx+K
J-K.X+Ky *K.
04-877-
9KLX (K+KX(+K, K. JK) 4x"
******
· © y = y + y = (C₁ + C. x) * + *-***
I 基本法则。
=> 据r(x)形式, 选函数
0
x) 0
• J
若r(x)为
令为
ker
Cer*
1+67+9+0
(7+5)-0
(+)
A-3₤k
kx" (no. 1.-) KX" KX"
K.X+K₂
kcos wx
Basic rule 1 - co
修正方
cecatene
Y-CLC" +c++--
14- 200-50
© y = y₁ + y₁ = (C₁ + C₁x + x) *
2-
e
Kcoswx + Msinwx
ksinwx
y" + y = 20" + 3x
y-y-o wil
sum rule
ㅍ修正法则。
=>
先据基本 rule选yp形式
若选之罪与恰相同形式
則 y,
₤ x "X" until y yo I 1]
Ye'(A+Binx)
= A casα + Brinx
②求非齐次,令= ce*+ Kx+ Ka
- Ice+K
Y"-40"
1A: SLC-KX+K₁ = 26 +3x
SC- K.-.). K₂O
y-y₁-y-Acosx+B+ e +3
Ⅲ总和法则
=>
若r(x)为多种函数之组合
则令,为多种函数之 sum
(maybe Modification rule)

ページ5:

Modeling :
: electric circuit
V₁ = IR VIL V-a - ±√In
=
=
Ide
E(t) = E sin cot
V₁₂ + V₁ + V₁ = E
=
+
IR IL
+
=
I
All
I'L + I'R + = = E, w, cos wit
+
+
L
I" - IR IE₁ we cos
SIE, sin w.t
E。w。
cos wit
=
L
①求齐次解
I'R I
'
令r(x)=0
①求非
次解(待定俘数法)
I" +
。
L LC
I₁ = c, e ₁₁t + C₁₂ e ₁₁t
½ Yp (t)
a cos w.t + b sin w.t
a =
- E.S
R*+5°
b = E.R
R'+S
(S= WL-C)
重根: I₁ = (c₁ + c₁t) eft
相异 複数根: In=e-** (Acoswt+Bsin wt)
ex
R 100 LIH COIF. E⋅ 10 sin 3t
VR=20I VIde₁ = I'l
I" 20 I10 10-3 cosst
①求齐次解,令r(x)=0
A++ 20入 + 10 = 0
λ=-20±√400-40
①求非齐次解(待定俘数法
r(t) - 30 cos st
I, Kwist + Msinst
= -10 ± 3√10
2
Ip
-3 Ksinst +3 M cost
Icetoo
三
Fourier
analysis
Fourier 級数: (週期2元)
1,"9K cost 9 Msinst
(K+60 M)cos ³t + (M-60 K) sin st = 30 cos st
(K+60M = 30
K = 110
{K = 3
IM-60K = 0
Icostsin st
M
I = I + Icetion. Cetion + the costsinst
任何週期函数f(x)
都可用三角函展开
t
b, sin x + b₂
sin 2x++
f(x) = a + a, cosx + a₂ cos xx + + a, cos nx = a + [a cos nx +
.不同频率成分
b. sin nx
一册
+ b₁₂ sin nx]
a = 1
π
n=1
a₁ =
2元
·Sth fixs cos nx
dx
b₁ = ± √ fix sin nx dx
fox) 代入
a. +
(a cos nx + b₁₂ sin nx)] dx
=
E
E
°
偶函数曲
a. a bo
a.dx+
an cos nx dx
+
ba sinnx dx
f(x)
奇函数 xxb.
-0
a. dx = ax
a.
+
a.
[a.+2
adx
‚× | * = ± a. (π +^)) = A。
-x
(a. cos nx + b₁ sin nx)] cosnx dx
fixt
an cos nx cos nx+
=1 OB
(a cos nx + b₁₂ sin nx)] sin.nx dx
fial
Cos mx
sin nxx +
0
TIXELIXIR
sin nx cos nx dx
= a
* fixi & fourier BR Ŕ
n sin xx sínnx]] dx
=
bp
-π
fix]
sinn
(10(-1)) = 0)
fix in wax) (√).
* 奇函 一口() 日月偶)

ページ6:

Fourier 級数:
T = 2π-
T= 2L
(任意週期2L)
'
w= 2πf = 1
f = =
.
W = 2πf=
f(x) =
=
= a + Σ [a cos (^x) + b₁₂ sin (^x)]
原始讯号
Fourier 級数:(非週期
T = ∞
a =
#S, fixo da
-L
a = ± S., fix cos xx, dx
-L
f(x)
sin(x) dx
fint for
T-6-L
L-3
| α = [] fide + [[ +1
}]
α- + [[ con
---cos)
fix) + (in)cor) + (sin cos (+
.
8-76
f(x) = √
元
②
1-00
nπ
L
3
F (w) etwx dw
00
F(w) = √2 o f(x) e-twx dx
8-
fcal
#7 {F(w)}
• fexs e-twx dx
F(W) = TIT
e
-iwx dx
=>
-
-
cos wx + i sin wx
ta eiwx 2
Fourier 反转換: f(x)= F {F(w)} (頻手还原成讯号)
Fourier 正转換: F(w)={f(x)} (怀晋校换湄频平)
(x20. f(x) = e-ax
x<0. f(x)=0
a20,求F{F(w)}
F(W) = f(x) e
ee
e
-jwx
dx
• √2π - (ariw) e
-(a+iw)x
dx
=√π (atów)
-7wx
-(a+iw)x dx
== [ee]
== (-isinw) =
sin (sinct)
A Review
• • sin (a = b)
=
sin a cosb ± cosa sinb
.
• cos (a = b) =
Cosa cosb
sina sin b
+ sina cosb = [sin(a+b) + sin(a - b)]
.
S
S
=
cos nx. cosmx dx (nm)
=== m) x d x + 1/1 f ^ ;
cos (n+m)x
cos nx. cosmx dx
|
(n= m)
cos (nm) x dx = 0+0=0
1 dx + cos 2nx dx = π +0 = π
sin nx sin mx dx (n+m)
0-0
cosa
sinb = [sin(a+b) -
- sin (a-b)]
3 +9
Cosa
+ cos(a-b)]
•
S
.
· cosb = = [ cos(a + b) +
sina sinb = [cos(a+b) = cos(a+b)]
= = = =
nx
=
-
L cos nx dx six-0
n
.
•
L
sin nx dx
cos nx
= 0
n
.
•
=
cos (n+m) x dx = 0-0=0
= cos (nm) x dx - co
S sin nx. sin mx dx (n= m)
S
1 dx = cos 2nxdx = ñ - 0 = ♫
sin nx⋅ cos mx dx
=
奇
14
-π
±±" sin (n+m) dx +
n = m
sin (nm).dx = 0

ページ7:

正合
(x² + y²) dx-2xydy = 0
非正合
M
x+2y
ay
JM
ON
ap
sin x cos y dx + cos x sin y dy = 0
M
JM
ay
-- sin x sing JMJN
ǝ(sin x cosy)
=
ay
aN
a (cosx sing)
ay ax
= - sin x sin
ax
ax
x sing
論
ǝx
TRIXIDA
Ju
2 x
= M =
Ju =
Sinx cosy
- sin x cosy dx + lys
•F Fix e
R (x) =
(-207)
=
ay
F=e
-214x1
=e
交
→
u =
Les Cosy
liys
ap
x²+y
equal
x
au
= N = cos x sin
cosx sing
| (-105x108y + llys) = cosxsiny - l´ (y) = cos x siny
P
a Q
ax
au
= P = 1
ax
Ju
-2%
Q
=
ligs =
= 0 =
liys-c
=> 4 = cos cosy + C
ǝy
Sau = Ju+1dx + lys
a (x - 1 + ligi) = -1 + liye -
5>>
u = x
X
ay
0
) - C
u =
M
= 2x
ay
ax
34
= M = 2xy
equal
Jau = {+xydx+lıyı u=yx" + lıys
an
2X
au
入
=
ay
x+
=
ay
u =
活体+5/2℃为常数,样本52.5 % (Tu = 5715 年),求样本年代
=>
y x +
X + C x
b
可分离
kt
y = ky m ft + ky + z dy k de Lin lazzs-bene was shy, et c' ayaq.ch
=
5715 πk at y²+ y².
=>
=
e
k5115
5115k
` = 0·5 — 57 15 k = -lu² = k = -0.0001213
-0.000111 t
e
=0.525 =>
t = 5312 (4)
=
每分流入
to tak
1000 gal
已有100籮小
(105)
求其中盐量at anyt
可分离
每分流出
y= Salt inflow rate - salt outflow rate = 500 of y
dy
0.01 ly - 50001 =
4 - 5000
- dy = f = 0.01 dt =
Inly-
e
50001 = -001t+C
5000 = -0.01
y
y 100 = 100 = e' + 5000
=
e-4900
=>
=
-4900e
+5000
必
V₂-IR
求工隨士变化
V₁ - I'L
y
E. V VIR + I'L
linear
=
dt
Kit)
I-e-hfertdt +ce-h
=
eft Edicets of
串联放射性decay: "Mo
AAN 99m To
AA99 To
N₂+ N₂ = ^N,
Ź
It
= λ, N,
λ, N₁
ANN, et
-λt
(JEE)
(出)
p(t) = ^₁₂
r (t) ^, N, e
-at
h- >dt
Sp(t) dt = {^, dt = 7₁t
N₂ = eh Jefritidt + ce-h
NA NA Ne
^, N₁ et dt + cert
eht
et
CAT
AN。
1
(X-t
-λit
e
+Ce
N₁ =
N
^-^
= 0
t-o
N=0
代入
C =
N₂
10(e)
23-dy
22

ページ8:

y" +0.4y+9.04y = 0,
y(0) = 0,
y'(0) = 3.
y' = ^e^, y = e^
ae+0.4入
+
9.0f e
= 0
入+0.4入+9.04 20
A = 0.16-4×9.04 <0 办
complex
root
W-6=3
入=-0.4±10.16-4-9.04 = -0.2 ± 3 i
2
0.+x
y
=
e
(A cos 3x + B sin 3x),
y (0) - ( A + 0)-A-0
-0.2x
E
y=
= e
^ (Bsin 3x)
y
= (-0.2 e
) (B sin 3x) + e
(3B cos 3x)
0.2x
=B (-02 e
-0.27
sin 3x + 3e
1053X)
y(0) B (0+3)= 3B = 3
=> B=1
y" +3y' +2.25y=-10e-1.5x y(0) = 1,
y'(0) = 0
①h
令r(x)=
y" + 3y² + 2.25y = 0
X + 3入 + 2 25 = 0
=-3-110=-1
(重根
y = ex sin 3xx
=
+x
= • cos x) y² = 0
+
| j i dy = f (e" - / + cos x) dx + c
=
In 1x1 + sin x + C
-= ex- = ln 1x1 - 2 sinx + C²
@ y = y + y = (c. + C.x −5׳) e¨´
8(0)=(₁ =1
-*x
Y₁ = (c. + C₁x) ex
②求和
basic rule y = c.ex
y= 2cxe (-) cxe
modify
LO
Ex
1" = 2 0 0 + x + 1 - 1 ) 2 ( x * - *ce + + 2 cxe *** = (2C - 6CX+ + (x²) e*x
y" -
-*
117 (2(-6CX+ = (x²) e ²x + 3(4x- = (x) e² + (xe
2ce ** = -10 € +*
9, -5xe
一言人
-x
- 10 e
y=(Cox) e + (- ±) (1 + C₁₂X - 5x)e
y' (o) - 0 = C₁ - 1 = 0
y = (1 + 3 × - 5׳) e
x
*
y" + 2y +0.75y = 2 cosx-0.25 sin x + 0.09x, y(0) = 2.78,
y'(0) = -0.43
-x
oth
令r(x)=
-y" + 2y² + 0.75y = o
②求
↑+ 2入+0.75.0
-214-475
2
Ja - C. e²±² + C. etx
Pasic rule ₤y - K cosx + Msinx + kx + k。
y - Ksinx + Mcosx + k₁
y=Kcosx-Msinx
(-Kox - Msinx) + (- Ksinx + M₁os x + k₁)
℗ y = y₁ - Y₁ = ce* + ce** + sin x + oπx - 032
UP
• t c₁ C₂
y (0) 2.78 C₁ + C₂+ 0.32
g² (0) = − 0 + 3 = ¯ ± 0, - ±0, +1 +0-12
C₁ = 31C-0
y = 3.1 e + sin x +0.12% - 0-32
+
0.75 (Kios x + Msin x +k₁x+k.) =
20051-0-25 sing4009x
⇒ K = 0 M² T
k=0·12-k=-032
y = sin x +0.12% -0.32

ページ9:

Find the current I(t)
© I L + IR +
0.1 [+ 11
diff
R= 110L01H
C⚫ 10th F
Q (0) = 0 1 (0) 0 E (t) = 110 sin 337 t
√ I de
+ 100
- Esin wit
of Ide
=/10 Sin 337 t
0. | I" + || [ + 100 [ = 110 cos 337 1
• * I h 令(x)=0
01入+11入+ 100 = 0
λ =
=
-11 ± √ 14-40
• P
0.2
+ C₂e
=7-10
a = -100
-100%
I, K cos 337 t + M sin 337t
④求C.C.
[(t) =
dQ
t
<
fIardt - Qus 1 1 5
0.1 I + 11 I + 100 Qit) = 10 sin 337t
0 | (0) + 10 + 0
= 0
[ (0) = C + C₁ - 2.71=0
I'(0) = -10C - 100 C₁ +0.796-337 = 0
C₁ = -0.323
=>
C₁ = 3.033
f(x) =
S = wL-C = 37.7-0.3=374
I = -0.323 e 10+ + 3.033 e
-loot
K-110374
112+374
- 2.71 cos 3377 + 0.796 sin 337t
= -27/
M
3
110 11
11+37-4
=
0.796
= −2.71 cos 337 ±+ + 0.7 9 6 s in 3 3 77
-k if
-T<x<0
and
f(x+2)= f(x)
k
if
0 < x <π
a
an
- —
-π
f(x)dx
= O
f" fexico sax dx
-π
-
½ [ -k½ sinnx" + k = v
b. - ½
S" fexas in n x dx
71
-π
=
夺函:
偶函
"
y²+ 8y + 7y = 0
A+81 +
= 0
64 -4.720 =>
相异實根
sinn
= 0
-2x
入 = -7
-
=
ncosnx
Cosnx
n
=
2-(1-cosmū)
it nodd
→ cosnit =
if even
4k
4k
by
b₂ = 0,
ba
TT
b40,
by
3πT
4k
5πT
→ 1-cosnπ
-1%
= 3e + 2x
y" + 4y ' + 4y =
D
f
(x) = 0
⇒
+
1+ 4y = 0
入²+4入 + 4入 = 0
16 - 4.4 = 0
⇒重恨
A=-2(重根)
-2x
f(x) = ±² (sin x + ¦ sín³x + ½ sinsx
71
-k if -2<x<0
|f(x)=
p = 2L = 4.
L 2
k if 0<x<2
a.
an
偶函
=
nπx
+--)
X
奇函
b. - ± [ - k sint ^*^*^) dx + = ['k sin( ^*^*^) dx
(
J₁ = (C,+C.X) e
Jp X²C e™ + K₁X²+ K‚ X + K.
J₁ = (2 x c − 2 X³ c) e
一%
+ · 2K₂X + K₁
y = 2ce-4xce* - 4xce* +4 x cc " + 4 ce™*x + 2K₂
=
(4x³c - 8xC + 6C) e™+2K₂
=
nд
2k (1- Cosnπ)
(-1. if nodd
159
→ cost
-> I-COST
1. if even
4k
4k
b₁ =
x2b₂ = 0.
b3
x2b4=0, bs
3π
4k
x2
5πT
|
fix) - 4 (sin x + + sin xxx + + sin xx ...
π
2
6ce + 4KX + 8K.x + 4K,X + 2K, +4K, +4K. - 30*+ 2x'
CIKK,--||K. - ½
3) • y = (C₁ + C₁x) e
-2x
-2x
+
3
+
e
+
TX
- x +
2
4

ページ10:

大
有一個 RLC 電路圖,如下圖所示,電阻器的電阻大小R-202(歐姆)。電容器的電容
C-10°F(法拉第),電感器的電感大小L-1.0H(亨利),當給予一電壓函數為E(t)=
小
2sin(5),當10時,電流與電荷皆為零,請求出此電路之電流隨時間之函數?
V₁ = I √₁₂ = 201, Ve= o'o, fide
I + 20] + 100 [Ide = 2 sin St
I + 20 ] + 100 [ = 2.5 cosst
• xk Iz
rit)=0
1 +20 + 100=0
入 = -10 (重根)
I-(C₁ct) e
R=200
• I
r(t) = 10 cos 5t, & I₁ = K cos 5t + Msinst
-10-15
5-5-20--15
K =
0.24
20+15
10 20
M-
0.32
20°+ 15°
= I₁ = 0.24 cosst + 0.323inst
-10t
© I = (C₁ + C₂t) C + 0.24 cost + 0.32 sinst
k
C=10 F
E(t)
10
=
3L=10H
Q (t)
I + 20 + 100 Q (t) = 2 sin st
[ 10)+0+0=0
I (0) = C, +0.24 = 0
I'(0) = -10 C₁ + C₁₂+0-32=0
-10t
1-(-0.24-2721) e
SC₁ = -0.24
C₁=-2·72
+0.44 cos 5t +0.32 Sinst
f(x+6) = f(x)
f(x+2) = f(x)
-πL
奇函
aan 18
ro
3
+3
b. - ½ S² -R sin(^^) dx + ≤ S R sm² ) dx
+
3
T= 6=2L L = 3
nx
f ^ ^
sin
x + y = π
y=π-x
听
" I
a. - ½ | ƒœx dx • ☆ [*< x-x) dx - F
+(x)
讯
10
nπ X
3
COS
+
LOS
RA
3
an -
元
f(x)cos x
=
(π-x)cosnx
-π
-3
odd
=
(π-x)(Sinnx) -
sinn x
→
nπ
even
=
=
= 0
f(x)=-47
sinx
- sin 3x + = sin 5x +-)
4
>
a =
if n
Godd
even
n
Cosnr
12, 9 = 0
π
cos nx
[ <- d x ) ]
2
a₁₂ =
=
1-cont
dv
(1 - Los nπ)
T-X
F(w)
Fourier 转换
fixie- wx dx
Te
dx
=
e
√2π
一直
+
- J'w lewis - elwa)
√272-iw
= √ET = iw (+ 2) sin (WIC))
Sin wit
√2πT
W
b₂ =
=
+(x) sin nx
da
(π-x) sinnx
元
=
dv
cosnx dz
-dx
sin nx
R-X
sinnxda
cosnx
-)-
-cosnx
n
T-
-dx
cosnx
sinni
+
sinnx
nπ
n
nπ
n) = h
f(x)=(sin x + sin 3x
sin 5x
25
)
(sin x + sinzx
Sinsx
3