2: ﺺﺨﻠﻣ 1: ﺺﺨﻠﻣ ﺔﯿﻠﺿﺎﻔﺘﻟا تﻻدﺎﻌﻤﻟا

1 ﺺﺨﻠﻣ

ﻦﻜﯿﻟ

:

و a ﯿﯿﻘﯿﻘﺣ ﻦﯾدﺪﻋ b ﻦ

.ﻦﯿﻣﺪﻌﻨﻣ ﺮﯿﻏ

:ﺔﯿﻠﺿﺎﻔﺘﻟا ﺔﻟدﺎﻌﻤﻟا لﻮﻠﺣ y¢ =ay +b

ﺔﯾدﺪﻌﻟا لاوﺪﻟا ﻲھ

ﻰﻠﻋ ﺔﻓﺮﻌﻤﻟا

¡ :ﻲﻠﯾ ﺎﻤﺑ

ax b

x ke -a a ﺚﯿﺣ k Ρ .

2 ﺺﺨﻠﻣ

:ﺔﯿﻠﺿﺎﻔﺘﻟا ﺔﻟدﺎﻌﻤﻟا ﻦﻜﺘﻟ

:

( )

^{E y¢¢+ay¢+by =0}

ةﺰﯿﻤﻤﻟا ﺎﮭﺘﻟدﺎﻌﻣ و

2 0

r + + =ar b ﺚﯿﺣ

و a نادﺪﻋb

.نﺎﯿﻘﯿﻘﺣ

§ ﻦﯿﻔﻠﺘﺨﻣ ﻦﯿﯿﻘﯿﻘﺣ ﻦﯿﻠﺣ ﻞﺒﻘﺗ ةﺰﯿﻤﻤﻟا ﺔﻟدﺎﻌﻤﻟا ﺖﻧﺎﻛ اذإ r1

2و r ,

ﺔﯿﻠﺿﺎﻔﺘﻟا ﺔﻟدﺎﻌﻤﻟا لﻮﻠﺣ نﺎﻓ

( )

^E

ﺔﻓﺮﻐﻤﻟا لاوﺪﻟا ﻲھ

¡ ﺎﻤﺑ ﻰﻠﻋ

:ﻲﻠﯾ

1 2

r x r x

x aae +be aﺚﯿﺣ

bو .نﺎﯿﻘﯿﻘﺣ نادﺪﻋ

§ جودﺰﻣ ﻲﻘﯿﻘﺣ ﻞﺣ ةﺰﯿﻤﻤﻟا ﺔﻟدﺎﻌﻤﻠﻟ ﺖﻧﺎﻛ اذإ r0

لﻮﻠﺣ نﺎﻓ ,

ﺔﯿﻠﺿﺎﻔﺘﻟا ﺔﻟدﺎﻌﻤﻟا

( )

^E

ﺔﻓﺮﻐﻤﻟا لاوﺪﻟا ﻲھ

¡ :ﻲﻠﯾ ﺎﻤﺑ ﻰﻠﻋ

( )

^{r x0}

xa ax+b e aﺚﯿﺣ

bو .نﺎﯿﻘﯿﻘﺣ نادﺪﻋ

§ ﻦﯿﻘﻓاﺮﺘﻣ ﻦﯿﯾﺪﻘﻋ ﻦﯿﻠﺣ ﻞﺒﻘﺗ ةﺰﯿﻤﻤﻟا ﺔﻟدﺎﻌﻤﻟا ﺖﻧﺎﻛ اذ

r1 = +p iq

2 و

r = -p iq ﺔﯿﻠﺿﺎﻔﺘﻟا ﺔﻟدﺎﻌﻤﻟا لﻮﻠﺣ نﺎﻓ ,

( )

^E

ﺔﻓﺮﻐﻤﻟا لاوﺪﻟا ﻲھ

¡ :ﻲﻠﯾ ﺎﻤﺑ ﻰﻠﻋ

(

^{cos sin}

)

x aepx a qx +b qx aﺚﯿﺣ

bو نادﺪﻋ

.نﺎﯿﻘﯿﻘﺣ

ﻦﯾﺮﻤﺗ 1

ﺮﺒﺘﻌﻧ

:

: ﺔﯿﻟﺎﺘﻟا ﺔﻟدﺎﻌﻤﻟا

( )

^{E :y¢ - =2 0}

1 ﺔﻟاﺪﻟا ﻞھ ( ﻰﻠﻋ ﺔﻓﺮﻌﻤﻟا f

¡ ﻲﻠﯾ ﺎﻤﺑ

( )

^{2 5} : f x = x+ ﻞﺣ

ﺔﻟدﺎﻌﻤﻠﻟ

( )

^E

؟

2 ﺎﻣ (

؟ تﻻدﺎﻌﻤﻟا هﺬھ ﻞﺜﻣو ﺔﯾدﺎﻋ ﺔﻟدﺎﻌﻣ ﻦﯿﺑ قﺮﻔﻟا ﻮھ 3 ﺔﻟدﺎﻌﻤﻠﻟ ﻞﺣ ﻦﻣ ﺮﺜﻛأ كﺎﻨھ ﻞھ (

( )

^E

؟

ﻦﯾﺮﻤﺗ 2

:ﺔﯿﻠﺿﺎﻔﺘﻟا ﺔﻟدﺎﻌﻤﻟا ﻞﺣ

: ( )

^{E : 2y¢ -4y- =3 0}

ﻦﯾﺮﻤﺗ 3 :

1 ﺔﻟدﺎﻌﻤﻟا ﻞﺣ( :ﺔﯿﻠﺿﺎﻔﺘﻟا

( )

^{:¨1 3 1 0}

E 2 y¢ + y- = 2 ﺔﻟاﺪﻟا دﺪﺣ( ﺔﯿﻠﺿﺎﻔﺘﻟا ﺔﻟدﺎﻌﻤﻟا ﻞﺣ f

( )

^E

: ﻖﻘﺤﺗ ﻲﺘﻟا

( )

^{0 2}

f¢ = -

ﻦﯾﺮﻤﺗ 4 :

1 :ﺔﯿﻠﺿﺎﻔﺘﻟا ﺔﻟدﺎﻌﻤﻟا ﻞﺣ(

( )

^{E :y¢¢-7y¢+12y=0}

2 ﺔﻟاﺪﻟا دﺪﺣ( ﺔﻟدﺎﻌﻤﻟا ﻞﺣf

( )

^E

ﻖﻘﺤﺗ ﻲﺘﻟا

( )

^{0 0}

f =

( )

^{0 1}و f ¢ =

ﻦﯾﺮﻤﺗ 5 :

: 1 :ﺔﯿﻠﺿﺎﻔﺘﻟا ﺔﻟدﺎﻌﻤﻟا ﻞﺣ (

( )

^{E :y¢¢-2y¢+ =y 0}

2 ﺔﻟاﺪﻟا دﺪﺣ( ﺔﻟدﺎﻌﻤﻟا ﻞﺣ f

( )

^E

ﻖﻘﺤﺗ ﻲﺘﻟا

( )

^{0 0}

f =

( )

^{0 1}و f¢ = .

ﻦﯾﺮﻤﺗ 6 :

1 :ﺔﯿﻠﺿﺎﻔﺘﻟا ﺔﻟدﺎﻌﻤﻟا ﻞﺣ(

( )

^{E :y¢¢-4y¢+13y=0}

2 ﺔﻟاﺪﻟا دﺪﺣ( ﺔﻟدﺎﻌﻤﻟا ﻞﺣ f

( )

^E

ﻖﻘﺤﺗ ﻲﺘﻟا

( )

^{0 0}

f =

( )

^{0 1}و f¢ = .

ﻦﯾﺮﻤﺗ 7

ﺔﯿﻠﺿﺎﻔﺘﻟا ﺔﻟدﺎﻌﻤﻟا ﻞﺣ

:

7 5

= -

'

y y

: ﺚﯿﺤﺑ 0 = -6 ( ) y

ﻦﯾﺮﻤﺗ 8

ﺔﯿﻠﺿﺎﻔﺘﻟا ﺔﻟدﺎﻌﻤﻟا ﻞﺣ

:

15 56 0

- + =

" '

y y y

: ﺚﯿﺤﺑ 0 =9 0 = -3 '( ) ; ( )

y y

ﻦﯾﺮﻤﺗ 9

ﻞﺣ

:

ﺔﯿﻠﺿﺎﻔﺘﻟا ﺔﻟدﺎﻌﻤﻟا

14 49 0

+ + =

" '

y y y

: ﺚﯿﺤﺑ 0 =6 0 = -3 '( ) ; ( )

y y

ﻦﯾﺮﻤﺗ 10

ﺔﯿﻠﺿﺎﻔﺘﻟا ﺔﻟدﺎﻌﻤﻟا ﻞﺣ

:

5 0

+ + 2 =

" '

y y y

: ﺚﯿﺤﺑ 0 =6 0 = -4 '( ) ; ( )

y y

ﻦﯾﺮﻤﺗ 11

:ﺔﯿﻠﺿﺎﻔﺘﻟا ﺔﻟدﺎﻌﻤﻟا ﺮﺒﺘﻌﻧ

:

2y¢+ + =4y 6 0

( )

^E

1 ﺔﯿﻠﺿﺎﻔﺘﻟا ﺔﻟدﺎﻌﻤﻟا ﻞﺣ .

( )

^E

2 ﺔﻟاﺪﻟا دﺪﺣ . ﺔﯿﻠﺿﺎﻔﺘﻟا ﺔﻟدﺎﻌﻤﻟا ﻞﺣ f

( )

^E

ﻲﺘﻟا

: طﺮﺸﻟا ﻖﻘﺤﺗ

( )

^{0 2}

f¢ =

ﻦﯾﺮﻤﺗ 12

:ﺔﯿﻠﺿﺎﻔﺘﻟا ﺔﻟدﺎﻌﻤﻟا ﺮﺒﺘﻌﻧ

:

1 2 1 0

3y¢ + y- =

( )

^E

1 ﺔﯿﻠﺿﺎﻔﺘﻟا ﺔﻟدﺎﻌﻤﻟا ﻞﺣ .

( )

^E

2 ﺔﻟاﺪﻟا دﺪﺣ . ﺔﯿﻠﺿﺎﻔﺘﻟا ﺔﻟدﺎﻌﻤﻟا ﻞﺣ f

( )

^E

: طﺮﺸﻟا ﻖﻘﺤﺗ ﻲﺘﻟا

( )

^{0 1}

f¢ = -

ﻦﯾﺮﻤﺗ 13

ﺔﻟدﺎﻌﻤﻟا ﺮﺒﺘﻌﻧ

:

:ﺔﯿﻠﺿﺎﻔﺘﻟا

5 6 0

y¢¢- y¢+ y=

( )

^E

1 ﺔﯿﻠﺿﺎﻔﺘﻟا ﺔﻟدﺎﻌﻤﻟا ﻞﺣ .

( )

^E

2 ﺔﻟاﺪﻟا دﺪﺣ . ﺔﯿﻠﺿﺎﻔﺘﻟا ﺔﻟدﺎﻌﻤﻟا ﻞﺣ f

( )

^E

: ﻦﯿطﺮﺸﻟا ﻖﻘﺤﺗ ﻲﺘﻟا

( )

^{0 2}

f =

( )

^{0 1} و f¢ =

ﻦﯾﺮﻤﺗ 14

ﺔﯿﻠﺿﺎﻔﺘﻟا ﺔﻟدﺎﻌﻤﻟا ﻞﺣ

:

( )

^{E :y¢¢-2 2y¢+2y =0}

ﻞﺤﻟا دﺪﺣ ﺔﯿﻠﺿﺎﻔﺘﻟا ﺔﻟدﺎﻌﻤﻠﻟ f

( )

^E

ﻦﯿطﺮﺸﻟا ﻖﻘﺤﯾ يﺬﻟا

( )

^{0 1}

f =

( )

^{0 0}و f ¢ = .

ﻦﯾﺮﻤﺗ 15

: ﺔﯿﻟﺎﺘﻟا ﺔﯿﻠﺿﺎﻔﺘﻟا تﻻدﺎﻌﻤﻟا ﻞﺣ

:

1 (

2 0

y¢¢+ y¢+ =y 2

(

4 8 0

y¢¢+ y¢+ y=

3 (

4 2 0

y¢¢- y¢+ y= 4

(

4 4 0

y¢¢- y¢+ y=

5 (

4 0

y¢¢ - y= 6

( 16 0 y¢¢ + y=

ﺔﯿﻠﺿﺎﻔﺘﻟا تﻻدﺎﻌﻤﻟا

ﻯﻭﺗﺳﻣﻟﺍ ﻙﺎﺑ ﺔﻳﻧﺎﺛﻟﺍ :

ﻳﺑﻳﺭﺟﺗ ﻡﻭﻠﻋ ﺔ

ﺔﻳﺋﺎﻳﺯﻳﻓ ﻡﻭﻠﻋ ﻙﺎﺑ ﺔﻳﻧﺎﺛﻟﺍ

:ﺫﺎﺗﺳﻷﺍ ﻭﻣﻠﺑ ﺩﻳﺷﺭ

:ﺔﻳﻠﻳﻫﺄﺗﻟﺍ ﺔﻳﻭﻧﺎﺛﻟﺍ

ﷲ ﺩﺑﻋ ﻱﻻﻭﻣ ﺭﻳﻣﻷﺍ