2301116 ก II

สมการเชิงผลตาง 1

2301116 ก II

ก

ก 2

ก

!"ก #"$%

!"ก #"$"

&!" ' (!"ก

ก &(ก)ก "')

ก &(ก)ก

ก 3

ก

*# y_x+"& ,!"** x +" x = 0, 1, 2, ... ก y_x+1–y_x

- "$%!" y_x+"ก. / - !" y_x !$#-∆y_x

∆y_x = y_x+1 - y_x

0# y₀ = 3, y₁ = 5, y₂ = 9, y₃ = 17, ... (1$#

∆y₀ = y₁ - y₀ = 2

∆y₁ = y₂ - y₁ = 4

∆y₂ = y₃ - y₂ = 8

!" ก "$"!" y_x

∆²y_x = ∆y_x+1 - ∆y_x

ก 4

ก "$

0# y₀ = 3, y₁ = 5, y₂ = 9, y₃ = 17, ... (1$#

∆²y₀ = ∆y₁ - ∆y₀ = 4 – 2 = 2

∆²y₁ = ∆y₂ - ∆y₁ = 8 – 4 = 4

"$!%.1& 1$#* "$-ก

0# F &234ก)!" x, y_x, y_x+1, ..., y_x+n+" n &2 -5-ก กก

F(x, y_x, y_x+1, ..., y_x+n) = 0

- ก y_x+1 + 4y_x = 3 y_x+1 + 8y_x = 2^x

ก 5

ก #"$%

* ∆y_x + ay_x = g(x)

กก #"$%

' , ก ∆y_x + 2y_x = 0

- -1&!"ก +" y_x = CA^x

+" A, C &2 - !" A $

∆y_x = y_x+1 - y_x = CA^x+1 - CA^x

!" ∆y_x 1$#

CA^x(A – 1) – 2CA^x = 0

ก 6

1$#- A = 3$. -1&+"

y_x = C3^x

1. -1&!"ก

y_x+1 + 5y_x = 0

!"ก

ก 7

ก -1&

ก -1&!"ก ∆y_x + ay_x = g(x) &(ก"$#- !."

!. 6 -1&!"ก

∆y_x + ay_x = 0 -1&+" y_x^H

!. 7 ' (!"ก

∆y_x + ay_x = g(x) ' (+" y_x^P

!. 8 ,)+"

y_x = y_x^H + y_x^P

ก 8

&!" g(x)

+" g(x) &234ก)' y_x^P &234ก)' $ก$-ก g(x)

0# g(x) = x² y_x^P &234ก)' $ก"$#- +"*# y_x^P = c₀ + c₁x + c₂x² #- c₀,c₁, c₂

+" g(x) &234ก)!.ก g(x) = B^x (1$#- y_x^H = CA^x

0# B ≠ A *# y_x^P = tB^x '#" t

0# B = A *# y_x^P = txB^x '#" t

+" g(x) &2,!"34ก)' ก34ก)!.ก g(x) = (x - 4)B^x (1$#- y_x^H = CA^x

0# B ≠ A *# y_x^P = (c₀ + c₁x)B^x

0# B = A *# y_x^P = (c₀ + c₁x)xB^x

ก 9

&!" g(x)

+" g(x) "*&-!"') / g(x) = g₁(x) + g₂(x)

0# y_x^P +" ' (!"ก ∆y_x + ay_x = g₁(x) ( y_x^Q+"

' (!"ก ∆y_x + ay_x = g₂(x) (1$#-

y_x^P + y_x^Q+" !"ก ∆y_x + ay_x = g₁(x) + g₂(x)

2. -1&!"ก

y_x+1- 8y_x = -6 – 7x

y_x+1- 2y_x = 6(5^x)

y_x+1- 2y_x = 3(2^x)

y_x+1- 8y_x = -6 – 7x + x8^-x

ก 10

ก &(ก)ก "')

-" - ')(- "- (x) ("$! *%&$ )

(y) !"# $%ก $"1&.

-# ! 1$#*%&$ ) +". -( 13 $ +"# -( 1 .;"$! "&$ )+" 5000

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d y d x=1

2



^p3y

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ก 11

ก "')*& f '(t) = k f(t)

ก "')"*& f '(t) = k f(t) $ f(t) &234ก)!%.

ก- ก$ก " ก <+"" " !"- ก

!"-0กก ก $" $"ก.#"+" &2#

!"ก "')+" f'(t) = k f(t)

ln f(t) = k t + c₁ f(t) = c e^kt 1

ftd f tk dt=0

∫

¹

ftd f t

∫

^{k dt=0}

ก 12

-" ก "')*& f '(t) = k f(t)

1. & ,!" " 90 ก +"" p(t) +"- 1& t &=

- ')ก$ก

- #"*#- 10% *# $ก -& ,+"'%

$-!"& ,&4 $ - .(*#- 1 &= '+"*#& ,$

+" 1/3 !"& ,&4

d

d t Pt=k Pt

ก 13

-" ก "')*& f '(t) = k f(t)

2. $#> CPI (f(t)) !"?"ก , &= t ก .@. 1980 &=

"$#"กก "')

f '(t) = 0.12 f(t) f(0) = 100

- *&= .@. 2007 &=$#> CPI !"?"ก ( ก 1

ก 14

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1. -&( กก p(t) ก&= '.@. 2543 ก "$#"ก ก "') p'(t) = k p(t) $&= '.@. 2534 ก&( ก 5.4

'# (*&= 2538 &( ก 6 '#

p(t) (&( กก*&= 2550

2. *# p(t)+" -' (*+."+"+"- 1& t กก $"

'- p(t) "$#"กก "') p'(t) = 0.55 p(t) !"

p(t) $ - +"#' (. - 10000 A)

ก 15

ก &(ก)ก

-" ก &(ก)ก ก*#ก$ @;?@ ) ก $*# S_t+"""!"&(@*&= t ก&4

I_t+"*#!"&(@*&= t ก&4

Y_t+" 1$#!"&(@*&= t ก&4

*#"" ( 1$#- ') ก

S_t+1 = 2Y_t+1, I_t+1 = 4(Y_t+1-Y_t), S_t+1 = I_t+1, Y₀ = k

+" k &2 - กก- @) ก S_t, I_t ( Y_t'#".&

-#> -("" > -(ก (> -( 1$#!"&(@

ก 16

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1. ก $*# C_t +" -*ก >.&(@*&= t ก&4

Y_t+" 1$#.&(@*&= t ก&4 I_t +"ก .&(@*

&= t ก&4 $- ')$ก "1&.

C_t+1 = 2Y_t + 5, Y_t = C_t + I_t, I_t = Y_t/2

ก C_t, Y_t( I_t

2. ก $*# C_t, Y_t( I_t *!#" (1) - ')$ก "

1&. Y_t = C_t + I_t, C_t = α + βY_t, Y_t+1 – Y_t = α I_t, Y₀ = k₁$ k₁, α > 0

( 0 < β < 1 ก !" Y_t