.1 LTI LTI.1 LTI y (n) () + a n-1 y (n-1) () + + a 1 y (1) () + a 0 y () = b m f (m) () + b m-1 f (m-1) () + + b 1 f (1) () + b 0 f () -

.1 LTI.3 0-0+.4. -1

.1 LTI LTI.1 LTI y (n) () + a n-1 y (n-1) () + + a 1 y (1) () + a 0 y () = b m f (m) () + b m-1 f (m-1) () + + b 1 f (1) () + b 0 f () -

.1 LTI y()( ) = y h ()( ) + y p ()( y (n) +a n-1 y (n-1) + +a 1 y (1) ()+a 0 y()=0 y h () P43-1- f() y () + 5y () + 6y() = f() 1 f() = e - 0y(0)=y (0)= -1 f() = e - 0y(0)= 1y (0)=0-3

.1 LTI : (1) + 5+ 6 = 0 1 = = 3 y h () = C 1 e + C e 3 - f() = e y p () = Pe Pe + 5( Pe ) + 6Pe = e P=1 y p () = e y() = y h () + y p () = C 1 e + C e 3 + e C 1,C y(0) = C 1 +C + 1 = y (0) = C 1 3C 1= 1 C 1 = 3 C = y() = 3e e 3 + e, 0-4

f()=e y p () = (P 1 + P 0 )e P 1 e - = e P 1 = 1 P 0 y()= C 1 e + C e 3 + e + P 0 e = (C 1 +P 0 )e +C e 3 + e y(0) = (C 1 +P 0 ) + C =1 y (0)= (C 1 +P 0 ) 3C +1=0 C 1 + P 0 =,C = 1 y() = e e 3 + e, 0 C 1 +P 0 = C 1 P 0-5.1 LTI

.1 LTI 0-0+ f() =0 C i = 0 + y (j) (0+) (j=0,1, n- 1) y (j) (0+) =0- y (j) (0-) y (j) (0-) y (j) (0+) -6

.1 LTI y () + 3y () + y() = f () + 6f() y(0-)=y (0-)= 0f()=() y(0 + )y (0 + ) f()=() y () + 3y () + y() = () + 6() 1 =0-0-<<0 + () () y () y ()= 0 y (0+)y (0-) y () y () () y () y()=0 y(0+) = y(0-) = -7

.1 LTI (1) 0+ 0-8 y' '( ) d + 3 0+ 0 y'( ) d + 0+ 0 y( ) d = 0+ 0 δ ( ) d + 6 0+ 0 ε( ) d [0-0 + ] y()=0 0+ 0+ y( ) d = 0, ε( ) d = 0 0 0 [y (0 + ) y (0-)] + 3[y(0 + ) y(0-)]= y(0+) = y(0-)= y (0 + ) y (0-) = y (0+) = y (0-) + = y() =0

.1 LTI y() = y x () + y f (), =0 f() y (j) x (0+), y (j) f (0+) (j = 01 n-1) y (j) (0-)= y (j) x (0-)+ y (j) f (0-) y (j) (0+)= y (j) x (0+)+ y (j) f (0+) y (j) x (0 + )= y (j) x (0 - ) = y (j) (0 - ) =0- y (j) f (0-)=0 y (j) f (0+) -9

.1 LTI y () + 3y () + y() = f () + 6f() y(0-)=y (0-)=0f()=() 1 y x () 0 y x () y x () + 3y x () + y x () = 0 y x (0+)= y x (0-)= y(0-)= y x (0+)= y x (0-)= y (0-)=0 1 y x () = C x1 e + C x e C x1 =4,C x = y x () = 4e e, > 0-10 10

.1 LTI y f () y f () + 3y f () + y f () = () + 6() y f (0-) = y f (0-) = 0 () y f () () y f () y f (0+)y f (0-) y f () = 0 y f (0+) = y f (0-) = 0 0+ 0+ [y f (0+)- y f (0-)]+ 3[y f (0+)- y f (0-)]+ y f ( ) d = + 6 0 ε( ) d 0 y f (0+)= y f (0-)= >0 y f () + 3y f () + y f () = 6 C f1 e - + C f e - 3 y f ()=C f1 e - + C f e - + 3 y f ()= 4e - + e - + 3 0-11 11

..3 () h()h()=t[{0},()] 1 y ()+5y ()+6y()=f() h() h() h () + 5h () + 6h() = () h (0-) = h(0-) = 0 h (0+)h(0+) -1 1

. () h () ()h () ()h (0+)h (0-)h()=0 h(0+)=h(0-) [h (0+) - h (0-)] + 5[h(0+) - h(0-)] + 6 h( ) d = 1 h(0+)= h(0-) h(0+)=h(0-)=0, h (0+) =1 + h (0-) = 1 >0 h () + 5h () + 6h() = 0 --3 h()=(c 1 e - + C e -3 )() C 1 =1,C =-1, h()=( e - -e -3 )() 0 + 0-13 13

. y ()+5y ()+6y()= f () + f () + 3f() h() h() h () + 5h () + 6h() = ()+ ()+3() (1) h (0-) = h(0-) = 0 h (0+)h(0+) h() () h() = a() + p 1 () [p i () () ] h () = a () + b() + p () h () = a () + b () + c()+ p 3 () (1) -14 14

. a () + b ()+ c() + p 3 () + 5[a () + b() + p () ] + 6[a() + p 1 () ] = ()+ ()+3() a ()+(b+5a) ()+(c +5b+6a)() + p 3 ()+5 p ()+6 p 1 () = () + () + 3() () a =1 b = - 3c = 1 h() = () + p 1 () h () = () - 3() + p () 3 h () = () - 3 () + 1()+ p 3 () 4 (3)0-0+ h(0+) h(0-) = 3 (4)0-0+ h (0+) h (0-) =1 h(0+) = 3h (0+) =1-15 15

. >0 h () + 6h () + 5h() = 0 3 h()= C 1 e + C e 3 >0 h(0+) = 3h (0+) =1 C 1 =3C = 6, h()= 3e 6e 3, > 0 () h()= () + (3e 6e 3 )() g()= T [() {0}] () () g( ) = h( τ )dτ, h( ) = d g( ) d -16 16

.3.3 1. (1) 1 p() 0 (a) 0 (b) f1() A f 1 () =? p() A f1( ) = p( ) = A p( ) 1-17 17

.3 () 0 f(0), f(0) p() f(0) p() f( ) 1 f(), -1 0 1 p( - ) 3 0 f() p( - ) -1 f(-) p( +) f ( - ) p( + ) f ˆ ( ) = f ( n ) p( n ) n = lim 0 fˆ ( ) f() f( ) f ˆ( ) = f ( ) = f ( τ ) δ ( τ ) d τ -18 18

-19 19.3. f () h() () ( -) f ()( -) f ( τ ) δ ( τ ) d f () y f ( ) = f ( τ ) h( y f () LTI τ τ ) h() h( -) d τ f () h( -) f y f () ( τ ) h( τ ) d τ

.3 3. f 1 ()f () f = ( τ ) dτ ( ) f1( τ ) f f 1 ()f () f()= f 1 ()*f () y f ( ) = f ( τ ) h( τ ) d τ = f ( ) * h( ) -0 0

.3 f () = e,-<<)h() = (6e - 1)() y f () y f () = f () * h() τ ( τ ) = e [6e 1] ε( τ )dτ y < > ( -) = 0 f ( ) = = e e τ [6e (6e ( τ ) 3τ 1]d τ = )dτ e τ dτ (6e e 3τ e τ )dτ = e e 3τ e τ = e e 3 e = e -1 1

.3 f = ( τ ) dτ 1( ) * f ( ) f1( τ ) f 1 f 1 ()f () f () f ( ) f (-) 3 f 1 () f (-) 4 -

-3 3.3 f (),h() y f ()= h() * f () [] h()h() 0 f () f () f () f (-) f ( -) < 0, f ( -) f ( -) h() = 0 y f () = 0 0 1, f ( -) 1 3 y y f 1 1 ( f ) = τ dτ = 0 4 1 1 ( ) = τ dτ = 1 1 1 1 y f ( ) = τ dτ = + 1 3 4 f ( -) h() = 0 y f () = 0 1 4 f ( -τ ) + f (-τ ) 1 f ( τ ) 1 h ( τ ) 0-1 -1-1 h(τ )f ( -τ ) = 3 4 3 4 1 4 1 τ 0-1 1-1 3 yf ( ) τ 0 1 3 τ

.3 f 1( τ ) f 1 (-) f 1 (-) - τ f ( τ ) f 1 ()f () 1 f() = f ()* f 1 () f() = τ -1 1 3 f ) = f ( τ ) f ( τ ) dτ -1 ( 1 1 f 1 ()f 1 ( ) 3f 1 ( ) f 1 ( ) 4f 1 ( )f () 5 f() = 0 - f 1(-τ ) f (τ ) τ -4 4

.4.4 1. f 1 ()* f () =f ()* f 1 (). f 1 ()*[ f ()+ f 3 ()] =f 1 ()* f ()+ f 1 ()* f 3 () 3. [f 1 ()* f ()]* f 3 ()] =f 1 ()*[ f () * f 3 ()] -5 5

-6 6.4 1. f()*()=()*f() = f() δ ( ) * f ( ) δ ( τ ) f ( τ )dτ = f ( ) = f()*( 0 ) = f( 0 ). f()* () = f () δ '( ) * f ( ) = δ '( τ ) f ( τ )dτ = f()* (n) () = f (n) () 3. f()*() = f ( τ ) ε ( τ )dτ = () *() = () f f ( τ ) dτ '( )

.4 n n n d 1 1. [ f f ] f f n 1 ( ) * ( ) = * ( ) ( ) * n = 1 n d d f d ( ) d f d ( ) = [ (n) () *f 1 ()] * f () = f (n) 1 () * f (). = (n) () *[f 1 ()* f ()] [ f1( τ ) * f ( τ )]dτ = [ f1( τ ) dτ ]* f ( ) = f1( ) *[ f ( τ )dτ ] = () *[f 1 ()* f ()] = [() *f 1 ()] * f () = f ( 1) 1 () * f () 3. f 1 ( ) = 0f ( 1) () = 0 f 1 ()* f () = f 1 ()* f ( 1) () -7 7

.4 1f 1 () = 1f () = e () f 1 ()* f () f ()* f 1 ()= τ τ τ e ε ( τ )dτ = e dτ = e 0 f 1 ()* f () = f 1 ()* f ( 1) () = 0* f ( 1) () = 0 0 f 1 (), f () = e () f 1 ()* f () f 1 ()* f () = f 1 ()* f ( 1) () f 1 () =() ( ) = 1 ( 1) τ τ τ f ( ) = e ε( τ )dτ e dτ = ε( ) = e 0 ε ( ) = (1 e ) ε ( ) 0 f 1 ()* f ()=(1- e )() [1- e (-) ](-) 1 0 f 1() -8 8

.4 f() = f 1 ()* f () f 1 ( 1 )* f ( ) = f 1 ( 1 )* f () = f 1 ()* f ( 1 ) = f( 1 ) f 1 (), f () = e () f 1 ()* f () f 1 () =() ( ) f 1 ()* f ()= () * f () ( ) * f () 1 0 f 1() () * f ()= f (-1) () ( ) * f ()= f (-1) ( ) f 1 ()* f ()=(1- e )() [1- e (-) ](-) -9 9

.4 f 1 (), f () f 1 ()* f f 1 () = () ( 1) f () = (+1) ( 1) f 1 ()* f () = ()* (+1) ()* ( 1) ( 1)* (+1) ( 1)* ( 1) ()* () = () f 1 ()* f () = (+1) (+1) - ( 1) ( 1) () ( ) ( ) -1 0 1 f 1() 1 f () 0 1-30 30

.4 1 3-31 31

.5.5 LTI LTI y (n) () + a n-1 y (n-1) () + + a 1 y (1) () + a 0 y () = b m f (m) () + b m-1 f (m-1) () + + b 1 f (1) () + b 0 f () y()( ) = y h ()( ) + y p ()( -3 3

.5 y (n) +a n-1 y (n-1) + +a 1 y (1) ()+a 0 y()=0 y h () P78-4 y () + 5y () + 6y() = f() 1 f() = e - 0y(0)=y (0)= -1 f() = e - 0y(0)= 1y (0)=0-33 33

.5 : (1) + 5+ 6 = 0 1 = = 3 y h () = C 1 e + C e 3-4 f() = e y p () = Qe Qe + 5( Qe ) + 6Qe = e Q=1 y p () = e y() = y h () + y p () = C 1 e + C e 3 + e C 1,C y(0) = C 1 +C + 1 = y (0) = C 1 3C 1= 1 C 1 = 3 C = y() = 3e e 3 + e, 0-34 34

f()=e y p () = (Q 0 + Q 1 )e Q 1 e - = e Q 1 = 1 Q 0 y()= C 1 e + C e 3 + e + Q 0 e = (C 1 +Q 0 )e +C e 3 + e y(0) = (C 1 +Q 0 ) + C =1 y (0)= (C 1 +Q 0 ) 3C +1=0 C 1 + Q 0 =,C = 1 y() = e e 3 + e, 0 C 1 +P 0 = C 1 P 0-35 35.5

.5 0+ f() =0 C i = 0 + y (j) (0+) (j=0,1, n- 1) y (j) (0+) =0- y (j) (0-) y (j) (0-) y (j) (0+) -36 36

.5 y () + 3y () + y() = f () + 6f() y(0-)=y (0-)= 0f()=() y(0 + )y (0 + ) f()=() y () + 3y () + y() = () + 6() 1 =0-0-<<0 + () () y () y ()= 0 y (0+)y (0-) y () y () () y () y()=0 y(0+) = y(0-) = -37 37

.5 (1) 0+ 0-38 38 y' '( ) d + 3 0+ 0 y'( ) d + 0+ 0 y( ) d = 0+ 0 δ ( ) d + 6 0+ 0 ε( ) d [0-0 + ] y()=0 0+ 0+ y( ) d = 0, ε ( ) d = 0 0 0 [y (0 + ) y (0-)] + 3[y(0 + ) y(0-)]= y(0+) = y(0-)= y (0 + ) y (0-) = y (0+) = y (0-) + = y() =0

.5 y() = y x () + y f (), =0 f() y (j) x (0+), y (j) f (0+) (j = 01 n-1) y (j) (0-)= y (j) x (0-)+ y (j) f (0-) y (j) (0+)= y (j) x (0+)+ y (j) f (0+) y (j) x (0 + )= y (j) x (0 - ) = y (j) (0 - ) =0- y (j) f (0-)=0 y (j) f (0+) -39 39

-40 40.5 1 y () + 3y () + y() = f () + 6f() y(0-)=y (0-)=0f()=() 1 y x () 0 y x () y x () + 3y x () + y x () = 0 y x (0+)= y x (0-)= y(0-)= y x (0+)= y x (0-)= y (0-)=0 1 y x () = C 1 e + C e C 1 =4,C = y x () = 4e e, > 0

.5 y f () y f () + 3y f () + y f () = () + 6() y f (0-) = y f (0-) = 0 () y f () () y f () y f (0+)y f (0-) y f () = 0 y f (0+) = y f (0-) = 0 0+ 0+ [y f (0+)- y f (0-)]+ 3[y f (0+)- y f (0-)]+ y f ( ) d = + 6 0 ε ( ) d 0 y f (0+)= y f (0-)= >0 y f () + 3y f () + y f () = 6 D 1 e - + D e - 3 y f ()=D 1 e - + D e - + 3 y f ()= 4e - + e - + 3 0-41 41

-4 4.5 h() () h()=t[{0},()] y ()+5y ()+6y()=f() h() h() h () + 5h () + 6h() = () h (0-) = h(0-) = 0 h (0+)h(0+) () h () ()h () ()h (0+)h (0-)h()=0 h(0+)=h(0-) 0 + 0 h( ) d

.5 [0-0 + ] [h (0+) - h (0-)] + 5[h(0+) - h(0-)] + 6 0 = 1 + 0 h ( ) d h(0+)= h(0-) =0 h (0+) =1 + h (0-) = 1 >0 h () + 5h () + 6h() = 0 --3 h()=(c 1 e - + C e -3 )() C 1 =1,C =-1, h()=( e - -e -3 )() -43 43