a; b 2 R; a < b; f : [a; b] R! R y 2 R:

y : [a; b]! R; ( y (t) = f t; y(t) ; a t b; y(a) = y : f (t; y) 2 [a; b]r: f 2 C ([a; b]r): y 2 C [a; b]; y(a) = y ; f y ỹ ỹ y ; jy ỹ j ky ỹk [a; b]; f y; ( y (t) = p(t)y(t) + q(t); a t b; y(a) = y :

p q [a; b]; p; q 2 C [a; b]; Z y(t) = R t t a hy p(s) ds + q(s) R i s a p() d ds ; a t b; a Z y(t) = R t t a p(s) ds y + q(s) R t s p() d ds; a t b: a p = ; y (s) = q(s); [a; t]; a < t b; y(t) y(a) = R t a q(s) ds; y(t) = y(a) + Z t a q(s) ds; a t b: y (s) p(s)y(s) = q(s) R s a p() d y(s) = R s a p() d q(s): a t; q q(t) = ( y (t) = 2ty(t) + t; t 2 R; y() = : p(t) = 2t q(t) = t: a = y = ; y(t) y(t) = t 2h + Z y(t) = R t t h 2s ds + s R i s 2 d ds ; t 2 R: Z t i s s2 ds ; t 2 R:

:= s 2 ; Z t s s2 ds = 2 y(t) = t 2h 2 Z t 2 d = 2 t 2 ; t 2 i = t 2h 3 2 2 t 2 i ; y(t) = 2 3 t 2 ; t 2 R: y (t) = y(t) + 3t; t t: p(t) = /t q(t) = 3t: y() = c; c; y(t) t; y(t) = t hc + Z y(t) = R t t s hc ds + 3s R i s d ds ; t 2 R: Z t i 3s s ds = h Z t i c + 3ss ds = t t (c + t 3 ); y(t) = t 2 + c t ; t > ; c = c : t: y (t) = p(t)y(t) + q(t)[y(t)] ; a t b;

2 R; = = y = v(t) := [y(t)] ; v (t) = ( )[y(t)] y (t); [y(t)] v (t) = p(t)y(t) + q(t)[y(t)] ; v (t) = ( )p(t)v(t) + ( )q(t); a t b: v; y v(t) = [y(t)] : [y(t)] [y(t)] [a; b]; y(t) y (t) = y(t) + [y(t)] 2 ; a t b; [a; b]: = 2: y; y(t) = t 2 [a; b]; v(t) := [y(t)] 2 = /y(t); y(t) [a; b]; v (t) = [y(t)] 2 y (t); [y(t)] 2 v (t) = y(t) + [y(t)] 2 v (t) = y(t) +

v (t) = v(t) ; a t b: v(t) = c t + ; c 2 R; y(t) = c t + ; a t b; c c t + [a; b]; c t + ; t 2 [a; b]: c t + a b; [a; b]: c [a; b]; a; b; b; a: c [a; b]; a; b; b; a: y (t) = r(t) + p(t)y(t) + q(t)[y(t)] 2 ; a t b: q = r = = 2: y y(t) = y (t) + z(t) ;

z; z y t; y (t) = y (t) [z(t)] 2 z (t): y(t) y (t) y (t) h [z(t)] 2 z (t) = r(t) + p(t) y (t) + i h + q(t) y (t) + i 2; a t b; z(t) z(t) y (t)r(t) p(t)y (t) q(t)[y (t)] 2 [z(t)] 2 z (t) = p(t) z(t) + 2q(t)y (t) z(t) + q(t) [z(t)] 2 : y [z(t)] 2 z (t) = p(t) z(t) + 2q(t)y (t) z(t) + q(t) [z(t)] 2 ; a t b; [z(t)] 2 ; z (t) = p(t) + 2q(t)y (t) z(t) + q(t); a t b: z y y y (t) = t y(t) [y(t)] 2 ; t 2; 2 t y (t) = /t y

y(t) := t + z(t) ; z; z(t) [; 2]; y (t) = t 2 [z(t)] 2 z (t): y(t) y (t) t 2 [z(t)] 2 z (t) = t 2 t t + z(t) t + 2; t 2; z(t) [z(t)] 2 z (t) = 3 tz(t) [z(t)] 2 ; t 2; [z(t)] 2 ; z (t) 3 z(t) = ; t 2: t y (t) = /t z; '(t) ; ' '(t) z (t) 3 t '(t) z(t) = '(t) : ' ' (t) = 3/t; t; '(t) = 3 t; '(t) = /t 3 ; t 3 z(t) = t 3 :

t Z t z(t) = 3 t dt = 3 2t + c; 2 z(t) = ct 3 t 2 ; t 2; c; c 2 R: y(t) = t + ct 3 t ; t 2; 2 c: [; 2]; '(t) := ct 2 2 t 2 [; 2]: ' c c < c ; '() '(2) ' [; 2] [; 2]: c ; ' [; 2]; '() c > /2; '(2) c < /8: ' [; 2]; c < /8 c > /2: y y (t) = /t c ct 2 /2; y y (t) = g(t) f (y(t)) :

t y; y = g(t) f (y) dy dt = g(t) f (y) : f (y) dy = g(t) dt: Z Z f (y) dy = g(t) dt; F (y) = G(t) + c; c: y t; y(t): dy dy dt dx: y = g(t) f (y) : y t: f (y(t)) f (y(t)) I;

f (y(s))y (s) = g(s); s 2 I; a I a t Z t a f (y(s))y (s) ds = Z t a g(s) ds; t 2 I: y I = y(s): y(a) y(t); d = y (s) ds; Z y(t) f () d = Z t y(a) a g(s) ds; t 2 I: F G f g; F (y(t)) F (y(a)) = G(t) G(a); t 2 I: y a; G(t); G(a); F (y(a)) F y: y y: y a; F (y(a)) c F (y(t)) = G(t) G(a) + c; t 2 I; y; G(a); c: y(t) y (t) = t + t 3 ;

a t a; Z t a y(s) y (s) ds = Z y(t) y(a) d = Z t a Z t a (s + s 3 ) ds; (s + s 3 ) ds; y(t) y(a) = t 2 2 + t 4 4 a2 2 a4 4 : y a; c y(a) a2 2 a4 4 ; y(t) = t 2 2 + t 4 4 + c: y c; R y(t); y(t) = t 2 2 + t 4 4 + c ; y (t) = 3t 2 + 4t + 2 2[y(t) ] ; t ; y() = ;

y ; t: 2[y(s) ]y (s) = 3s 2 + 4s + 2 t; t; Z t 2[y(s) ]y (s) ds = = y(s); Z y(t) 2( ) d = Z t y() Z t (3s 2 + 4s + 2) ds; (3s 2 + 4s + 2) ds; [y(t)] 2 2y(t) [y()] 2 2y() = t 3 + 2t 2 + 2t: y() = ; [y(t)] 2 2y(t) = t 3 + 2t 2 + 2t + 3: y (t) = p t 3 + 2t 2 + 2t + 4; y 2 (t) = + p t 3 + 2t 2 + 2t + 4; t : y 2 (t) y 2 () = 3 y(t) = y (t); t : y (t) = ty(t)[y(t) 2] y(t) = y(t) = 2 y(t) 2 y (s) y(s)[y(s) 2] = s:

a a t; Z t := y(s); a y Z (s) t y(s)[y(s) 2] ds = s ds; a Z y(t) y(a) Z t ( 2) d = s ds: a ( 2) = 2 2 ; h Z y(t) Z y(t) 2 y(a) 2 d i Z t y(a) d = s ds; a t 2 jy(t) 2j jy(t)j jy(a) 2j jy(a)j = 2 2 2 a2 2 : jy(a)2j jy(a)j a 2 c; a y a; jy(t) 2j jy(t)j = t 2 + c; zc ; ˇ ˇy(t) 2 ˇ = t 2 + c; y(t) ˇ ˇ 2 ˇ = t 2 + c: y(t) ˇ ˇ 2 ˇ = zc t 2 ; y(t) 2 y(t) = C t 2 ;

C; y(t) = 2 + C t 2 C : C; y + C t 2 : y 2: C = ; y(t) = 2: y(t) = C; y (t) = g y(t) t f f (t; y) = f (t; y) 8; t; y 2 R: M N f; f (t; y) := M (t; y)/n (t; y); y (t) = f t; y(t) ; M (t; y) dt + N (t; y) dy = ; g(s) := f (; s); f (t; y) = f (t ; t y t ) = t f (; y t ) = f (; y t ):

v(t) := y(t)/t; t; y(t) = tv(t); y (t) = tv (t) + v(t); tv (t) = g v(t) v(t); g v(t) v(t) t; v (s) g v(s) v(s) = s : a a t a t Z t a v Z (s) t g v(s) v(s) ds = a s ds; = v(s); Z v(t) v(a) d = jtj jaj: g() G /[g() ]; G(v(t)) G(v(a)) = jtj jaj; t 2 I: y a; G v: v v: v a; G(v(a)) jaj c G(v(t)) = jtj + c; t 2 I;

v; v; y y(t) = tv(t): g v(t) v(t) v? 2 R g; g(v) = v; tv (t) = ; v v(t) = v? : y(t) = v? t y (t) = [y(t)]2 + 2ty(t) t 2 v(t) := y(t)/t v(t) + tv (t) = [v(t)] 2 + 2v(t) tv (t) = [v(t)] 2 + v(t): v 2 + v = v = v = : y(t) = y(t) = t v(t) v (s) v(s)[v(s) + ] = s :

a a t; Z t a v (s) v(s)[v(s) + ] ds = := v(s); Z v(t) v(a) ( + ) d = Z t Z t a a s ds; s ds: Z v(t) v(a) ( + ) = + ; Z v(t) Z d t v(a) + d = a s ds; jv(t)j jv(t) + j jv(a)j jv(a) + j = jtj jaj: jv(a)j jv(a) + j jaj jcj; c; a v a; jv(t)j jv(t) + j = jtj + jcj; v(t) ˇ ˇ = jctj; v(t) + v(t) v(t) + = ct; c: v(t) = y(t) = tv(t) ct ct ; y(t) = ct 2 ct c: c; y t /c:

y M (t; y(t)) (t) = N (t; y(t)) : f (t; y) @f @t (t; y) = M (t; y) @f (t; y) = N (t; y): @y M (t; y) dt + N (t; y) dy = : f d dt f (t; y(t)) = @f @f (t; y(t)) + @t @y (t; y(t)) y (t) = M (t; y(t)) + N (t; y(t)) y (t) = ; f (t; y(t)) f (t; y(t)) = c; c: y y; f f: M N f @ 2 f @t@y = @2 f @y@t ;

@M @y = @N @t : f Z f (t; y) = M (t; y) dt + g(y); g; @f = M: y; @f = N @t @y Z M y (t; y) dt + g (y) = N (t; y): M y N t ; Z N t (t; y) dt + g (y) = N (t; y) Z g (y) = N (t; y) N t (t; y) dt: y; Z h Z g(y) = N (t; y) i N t (t; y) dt dy + C; C: f M N; M N f; Z Z h Z i f (t; y) = M (t; y) dt + N (t; y) N t (t; y) dt dy + C: y(t) y (t) = t y(t) + 2y(t) :

M (t; y) := y N (t; y) := t y + 2y; @M @y (t; y) = @N @t (t; y) = y ; f (t; y) @f (t; y) = y @t @f @y (t; y) = ty + 2y: @f (t; y) = y H) @t f (t; y) = t y + g(y); g(y): y @f @y (t; y) = ty + 2y t y + g (y) = t y + 2y; g (y) = 2y; g(y) = y 2 + c; c: f (t; y) = t y + y 2 + c; c: t y(t) + [y(t)] 2 = c; c: ; y (t; y(t))m (t; y(t)) (t) = (t; y(t))n (t; y(t)) @(M ) @y = @(N ) @t

@M @y + M @ @y = @N @t + N @ @t N @ @t M @ = @M @y @y @N @t : t y; = = (t): d dt = @M @y t; y; = (t) @N @t N : (t) = R @M @N @y @t N dt : = = (y): @M d dy = @N @y @t M : y; t; = (y); (y) = R @M @N @y @t M dy : @M @y @N @t N y; = (t); @M @y @N @t M

t; = (y); y (t) = y(t) t 2 y(t) t : M (t; y) := y N (t; y) := t 2 y t; @M @y (t; y) = ; @N @t (t; y) = 2ty ; @M @y (t; y) @N @t ; @M @y @N @t M (t; y) = 2ty + 2 y = 2t + 2 y t; = (y) y: @M @y @N @t N (t; y) = 2ty + 2 t 2 y t = 2 t y; = (t) t: d dt (t) = 2 t (t) (t) = 2 t ; j(t)j = 2 jtj = t 2 (t) = t 2 : (t) = /t 2 : /t 2 ; y(t) t 2 y (t) = ; y(t) t

f = f (t; y) @f @t (t; y) = y t 2 @f @y ty (t; y) = : t @f @t (t; y) = y Z H) f (t; y) = t 2 @f @y (t; y) = t + g (y); t + g (y) = ty t y t 2 dt + g(y) = y t + g(y): g (y) = y; g(y) = 2 y2 + c; c: f (t; y) = y t + 2 y2 + c; y(t) y(t) t c: + 2 [y(t)]2 + c = ; Η γραμμική Δ.Ε. ανάγεται σε πλήρη. y (t) = p(t)y(t) + q(t) y M (t; y(t)) (t) = N (t; y(t)) M (t; y) = p(t)y q(t) N (t; y) = : p; p(t) = @M @y (t; y) @N @t (t; y) = ; p = h @M N (t; y) @y (t; y) @N i @t (t; y) = p(t)

t; (t) (t) = R [p(t)] dt = R p(t) dt ; R p(t) dt y (t) = p(t)y q(t) R p(t) dt = z M (t; y(t)) zn (t; y(t)) ; zm zn : f = f (t; y) @f @t (t; y) = zm (t; y) = p(t)y + q(t) R p(t) dt @f @y (t; y) = zn (t; y) = R p(t) dt : f (t; y) = R p(t) dt y + g(t) g: t Z g (t) = q(t) R p(t) dt ; g(t) = q(t) R t a p(s) ds dt; a p q f Z f (t; y) = R p(t) dt y q(t) R t a p(s) ds dt: f (t; y(t)) = C; Z R p(t) dt y(t) q(t) R t a p(s) ds dt = C; C: y(t) y(t) = R h Z p(t) dt C + q(t) R i t a p(s) ds dt ;

n n ( y (t) = A(t)y(t) + f (t); a t b; y(a) = y () : y : [a; b]! R n f : [a; b]! R n y (t); : : : ; y n (t) f (t); : : : ; f n (t); A : [a; b]! R n;n y (t) f (t) a (t) a 2 (t) : : : a n (t) y 2 (t) y(t) = B C @ : A ; f (t) = f 2 (t) B C @ : A ; A(t) = a 2 (t) a 22 (t) : : : a 2n (t) B C @ : : : A ; y n (t) f n (t) a n (t) a n2 (t) : : : a nn (t) t 2 [a; b]: ( y (t) = p(t)y(t) + q(t); a t b; y(a) = y ; p q [a; b]; y(t) = R t a p(s) ds y + Z t a q(s) R t s p() d ds; a t b:

A A: A(t) A(s) t s [a; b]; A(t) = p(t)a; p A 2 R n;n : A(t) = A; A t; ( y (t) = Ay(t) + f (t); a t b; y(a) = y () : ( y (t) = Ay(t); t 2 R; y() = y () ; a; t = s + a: y () ; y(t) = : ( y (t) = ay(t); t 2 R; y() = y ;

y(t) = at y ; y(t) = t y () ; t 2 R; 2 C; : y(t) = t y () H) y (t) = t y () ; y y (t) = Ay(t); t y () = A t y () = t Ay () ; Ay () = y () : y () y () = y(t) = t y () ; t 2 R; A y () y () A ; : : : ; m A x () ; : : : ; x (m) y () = c x () + + c m x (m) ; c ; : : : ; c m : y y(t) = c t x () + + c m mt x (m) ; t 2 R: y (t) = c t x () + + c m m mt x (m)

Ax (i) = i x (i) ; i = ; : : : ; m; Ay(t) = c t Ax () + + c m mt Ax (m) = c t x () + + c m mt m x (m) ; y (t) = Ay(t); t 2 R: x () ; : : : ; x (m) c ; : : : ; c m y () 2 R n A 2 R n;n : A n x () ; : : : ; x (n) 2 C n : A A x () ; : : : ; x (n) C n ; y () y () = c x () + + c n x (n) ; c i y () ; n n c i ; (x () ; : : : ; x (n) ); x (i) : y(t) = c t x () + + c n nt x (n) ; t 2 R; i A x (i) a; y(a) = y () ; y(t) = c (ta) x () + + c n n(ta) x (n) ; t 2 R: y () A: y () ; A

( y (t) = ay(t); t 2 R; y() = y ; y(t) = at y ; y(t) = ta y () ; t 2 R; A ; A: z ; z 2 C; A 2 C n;n ; = I n I n A := I n + A + A2 2! + + A` `! + = X `= A` `! : = I n ; ; ta ta = A ta : A B = A+B ; A B AB = BA: y() = A y () = y () = I n y () = y () : y (t) = ta y () = ta y () = A ta y () = Ay(t); t 2 R; a; y(a) = y () ; y(t) = (ta)a y () ; t 2 R:

ta y () ( y (t) = Ay(t) + f (t); t 2 R; y() = y () ; ta y () ; y(t) = ta v(t); t 2 R; v: y() = y () v() = y () : y (t) = ta v(t) = ta v(t) + ta v (t) = A ta v(t) + ta v (t) = Ay(t) + ta v (t); y (t) = Ay(t) + f (t) ta v (t) = f (t): sa v (s) = f (s) v (s) = sa f (s); t v() = y () ; v(t) y () = v(t) = y () + Z t Z t sa f (s) ds; sa f (s) ds; t 2 R: Z t y(t) = ta v(t) = ta y () + ta sa f (s) ds; y(t) = ta y () + Z t (ts)a f (s) ds; t 2 R;

y (t) = ay(t) + f (t): a; y(a) = y () ; y(t) = (ta)a y () + Z t a (ts)a f (s) ds; t 2 R: ta y () ta y () ta x; x 2 C n ; t 2 R: = x A : A I n I n ; ta x = ti n t(ain) x = t I n t(ain) x = t t(ain) x h = t I n x + t(a I n )x + t 2 i 2! (A I n) 2 x + ; (A I n )x = (A I n )`x = ; `; ta x = t x: y () A: = x (A I n ) m x = ; A m: ta x = t t(ai n) x = t h I n x + t(a I n )x + t 2 + t m i m! (A I n) m x + ; 2! (A I n) 2 x + + t m (m )! (A I n) m x

(A I n )`x = ; ` m; ta x = t h I n x + t(a I n )x + t 2 2! (A I n) 2 x + + t m (m )! (A I n) m x i : n A C n A ta y () = A A A C n A A m: x 2 C n (A I n ) m x = A : A m; m C n A: m A ; m (A I n )x = A

(A I n ) 2 x = A A; ; (A I n ) 3 x = A A; (A I n )`x = ; ` ; m : n x () ; : : : ; x (n) A: y ()

x () ; : : : ; x (n) ; y () = c x () + + c n x (n) ; y(t) y(t) = ta y () = c ta x () + + c n ta x (n) ; t 2 R: = = ta x (i) ; x (i) A; y(t) = ta y () : a; ' (i) (t) := ta x (i) ; t 2 R; i = ; : : : ; n; y (t) = Ay(t) y (t) = Ay(t); y () ' (i) ; i = ; : : : ; n: x (i) ; i = ; : : : ; n: x (i) ; i = ; : : : ; n; A ' (i) ; i = ; : : : ; n: 4 y B C (t) = Ay(t); t 2 R; A := @ 3 2 A: 2 p A p() = 3 + 2 2 + 5 6: 6 ; 2; 3; p; ; 2; 3: A = ; 2 = 3; 3 = 2: p p: p p:

= v 2 R 3 (A I 3 )v = (A I 3 )v = 4 B CB @ 3 A@ 2 2 v v 2 v 3 C A = ; v 2 + 4v 3 = 3v + v 2 v 3 = 2v + v 2 2v 3 = v + v 3 = ; v = v 3 ; v 2 = 4v 3 : v 3 ; v v 2 : B C v = @ 4A: ' () (t) = t B C @ 4A; t 2 R: 2 = 3 v 2 R 3 (A 2 I 3 )v = (A 3I 3 )v = 2 4 B CB @ 3 A@ 2 4 v v 2 v 3 C A = ; ƒ 2v v 2 + 4v 3 = 3v v 2 v 3 = 2v + v 2 4v 3 = v = v 3 ; v 2 = 2v 3 : B C v = @ 2A; ' (2) (t) = 3t B C @ 2A; t 2 R: : ƒ ;

3 = 2 v 2 R 3 (A 3 I 3 )v = (A + 2I 3 )v = 3 4 B CB @ 3 4 A@ 2 v v 2 v 3 C A = ; 3v v 2 + 4v 3 = 3v + 4v 2 v 3 = 2v + v 2 + v 3 = v = v 3 ; v 2 = v 3 : B C v = @ A; ' (3) (t) = 2t B C @ A; t 2 R: y(t) c t + c 2 3t c 3 2t y(t) = c t B C @ 4A + c 2 3t B C @ 2A + c 3 2t B C B @ A = @ 4c t + 2c 2 3t + c 3 2t C A; c t + c 2 3t + c 3 2t t 2 R; c ; c 2 c 3 : y(2) = (; 2; 3) T 2 R 3 : c t + c 2 3t c 3 2t B y(t) = @ 4c t + 2c 2 3t + c 3 2t C A; c t + c 2 3t + c 3 2t c ; c 2 ; c 3 c ; c 2 ; c 3 y(2) = (; 2; 3) T : ƒ ;

? c 2 + c 2 6 c 3 4 = ƒ 4c 2 + 2c 2 6 + c 3 4 = 2 : c 2 + c 2 6 + c 3 4 = 3 (?) 2c 2 6 = 4; c 2 = 2 6 : (?) c 2 c 3 4 = c 2 6 = 2 = 4c 2 + c 3 4 = 2 2c 2 6 = 2 4 = 2: ) c 2 c 3 4 = : 4c 2 + c 3 4 = 2 3c 2 = 3; c = 2 : c 3 4 = H) c 3 4 = 2 H) c 3 = 2 4 : c ; c 2 ; c 3 t2 + 2 3t6 2 2t+4 B y(t) = @ 4 t2 + 4 3t6 + 2 2t+4 C A t2 + 2 3t6 + 2 2t+4 y B C (t) = Ay(t); t 2 R; A := @ A: 2

p A p() = (2 )( ) 2 ; A = 2 2 = = 2 v 2 R 3 (A I 3 )v = (A 2I 3 )v = B CB @ A@ v v 2 v 3 C A = ; v = v 2 = v 3 v 3 v 3 = ; ' () (t) = 2t B C @ A; t 2 R; 2 = v 2 R 3 (A 2 I 3 )v = (A I 3 )v = v B CB C @ A@ v 2 A = ; v 2 = v 3 = v v v = ; ' (2) (t) = t B C @ A; t 2 R; 2 2 v 2 R 3 (AI 3 ) 2 v = (AI 3 )v : (AI 3 ) 2 = ; (A I 3 ) 2 v = v B CB C @ A@ v 2 A = ; v 3 v 3

v 3 = v v 2 v = v 2 = 2 : ' (3) (t) = t v + t(a I 2 )v = t B C @ A + t t B CB C @ A@ A = t B C @ A + t t B C @ A; t ' (3) (t) = t B C @ A; t 2 R; y(t) t y(t) = c 2t B C @ A + c 2 t B C @ A + c 3 t B C @ A; (c 2 + c 3 t) t B y(t) = @ c 3 t C A; t 2 R; c 2t c ; c 2 c 3 : x ; x 2 R; n n A := I n + A + A2 2! + + A` `! + = X A 2 R n;n : A A A; B 2 R n;n t; s 2 R `= A` `!

ta = ta : (t+s)a = ta sa : t(a+b) = ta tb ; A B AB = BA: A+B = A B ; A B A B t = A B = X `= A` `! X `= B` `! = X `X `= k= A`k (` k)! B k = k! X `= `! (A + B)` = A+B : ta ta = A + ta 2 + + t ` A`+ `! + = A I n + ta 2 + + t ` A` `! + ; ta = A ta : ( y (t) = Ay(t); a t b; y(a) = y () ; A = a ij i;j 2 =;:::;n Rn;n t y(t) y(t) = (ta)a y () ; a t b: y y (t) = (ta)a y () = ta aa y () = A ta aa y () = A (ta)a y () = Ay(t):

= I n ; y(a) = A y () = I n y () = y () : ( y (t) = Ay(t) + f (t); a t b; y(a) = y () : y(t) y(t) = (ta)a hy () + y(t) = (ta)a y () + Z t a Z t a i (sa)a f (s) ds ; a t b; (ts)a f (s) ds; a t b: y (t) = ay(t) y (t) = ay(t) + f (t); y(t) = (ta)a v(t); a t b; v(t): y () v(t) t: v(t): y(a) = v(a); v(a) = y () : (ta)a v (t) = f (t) A (ta)a v(t) + (ta)a v (t) = A (ta)a v(t) + f (t); v (t) = (ta)a f (t); a t b:

v; v s t; [a; t]; a t b; v(a) = y () ; v(t) = y () + Z t a (sa)a f (s) ds; a t b: (ta)a A: A (ta)a A v; v; A; A p; p() := (A I n ); A: = A A A a ij ; i j; a ii A i A = ( ; : : : ; n ): n y i (t) = iy i (t); A` = (` ; : : : ; ǹ); ` 2 N ; A = ; : : : ; n :

y(t) (ta) y () y(t) = 2(ta) y () 2 B C @ : A ; a t b; n(ta) y () n y () ; y () 2 ; : : : ; y () n y () : A A B; S A = S BS SAS = B A B A 2 = (S BS)(S BS) = S B 2 S A` = S B`S; ` 2 N : A = X `= A` `! = X `= S B`S `! = S X `= B` `! S; A = S B S: A = ( ; : : : ; n ) S S AS = : AS = S; A S n n A n A A n A S n; A:

A = S ; : : : ; n S : y(t) y(t) = S (ta) ; : : : ; n(ta) S y () ; a t b: A n S AS = ; y (t) = Ay(t) y (t) = SS y(t) S y (t) = S y(t); z(t) := S y(t); z (t) = z(t): z(a) = S y(a) = S y () =: z () : ( z (t) = z(t); a t b; z(a) = z () : n zi (t) = iz i (t); z i (a) = z () i ; z i (t) = i (ta) z () i ; i = ; : : : ; n; z(t) = (ta) ; : : : ; n(ta) z () ; a t b: z(t) S y(t) z () S y () ; = A A A m A m A m = :

A` = ; ` m; A` = A`m A m = A`m = : m; A m = ; A: A m n n A n; A m; A = m X `= y(t) y(t) = m X `= A` `! (t a)` A`y () ; a t b: `! A A = I n +M; M m: I n = I n ; I n nn I n n n M; A = I n+m = I n M = I n M = M ; m X A = `= M ` `! : y(t) m X y(t) = (ta) `= (t a)` M `y () ; a t b: `! n n A A J = (J ; J ; : : : ; J k ); J

J` ` ` J` = : : : : : : 2 C n`;n`; ` = ; : : : ; k; B C @ ` A ` n n S A = S JS: A J = J ; J ; : : : ; J k : J` J` = `I n` + M n` M n` 2 R n`;n` M n` = : : : : : : : B C @ A M n` M n` B @ x x 2 : x n` x n` x 2 x 3 = : ; C B C A @ x n` A x M n` x = ; n` x 2 C n`; M n` = : n` M n` n`:

M k n` M n` M 2 : : : : : : = ; : : : ; M n` n` B C @ A n` = : : : : : : ; M n` = : n` B C @ A Σχετικά με το γεγονός ότι ο πίνακας M n` είναι μηδενοδύναμος. M n` M n` = M n` n`; J = (J ; J ; : : : ; J k ) J ` = (J ` ; J ` ; : : : ; J ` k ); ` 2 N ; A = S J ; J ; : : : ; J k S: J ; J i ; i = ; : : : ; k; y(t) y(t) = B(t)y () ; a t b; B(t) 2 C n;n ; a t b; b ij (t) b ij (t) = mx `= p (i;j ) ` (t) `t ; a t b;

` A p (i;j ) ` (t) `: A v (ta)a v v; A ; y(t) (ta)a y () ; (ta)a : v; y () : y () v () ; : : : ; v (k) : t; 2 R; (ta)a = (ta)i n (ta)(ai n) = (ta) I n (ta)(ai n) = (ta) (ta)(ai n) ; v 2 C n ; (ta)a v = (ta) (t a)` v + (t a)(a I n )v + + (A I n )`v + : `! A: v (A I n )v = (ta)a v = (ta) v: A n y () A C n A: v 2 C n A ; (A I n ) v = : (A I n ) +`v = ; ` 2 N ; (A I n ) +`v = (A I n )`(A I n ) v = (A I n )` = : (ta)a v = (ta) v + (t a)(a I n )v + + (t a) (A I n ) v : ( )!

; A n n A n y () A: A k ; : : : ; k ; : : : ; k ; : : : ; k ; y () y () = v () + + v (k) ; v (i) A i ; i = ; : : : ; k: y(t) y(t) = kx (ta)a v (`) = `= kx `= k X `(ta) m= (t a) m (A I n ) m v (`) ; a t b: m! A: n n y (t) = Ay(t); y n A nn y (t) a a 2 : : : a n y 2 (t) y(t) = B C @ : A ; A := a 2 a 22 : : : a 2n B C @ : : : A : y n (t) a n a n2 : : : a nn

t 2 R y () 2 R n ; y(t ) = y () ; y () (t) y (2) (t) y () (t) + y (2) (t) y () (t) + y (2) (t) y () (t) V: V n; V = n: Διάσταση του χώρου λύσεων της. V n; V = n: V n i 2 f; : : : ; ng; ' (i) ( y (t) = Ay(t); t 2 R; y() = e (i) ; fe () ; : : : ; e (n) g R n ; e (i) j = ı ij ; i; j = ; : : : ; n: ' () ; : : : ; ' (n) c ' () + + c n ' (n) = ; c ; : : : ; c n ; c ' () () + + c n ' (n) () = ; c e () + + c n e (n) = ; e () ; : : : ; e (n) ; c = = c n = : y y ' () ; : : : ; ' (n) ; c ; : : : ; c n y(): z; z := c ' () + + c n ' (n) ;

z z() = c ' () () + + c n ' (n) () = c e () + + c n e (n) = y(); y: y ' () ; : : : ; ' (n) ; n y (t) = ay(t) y(t) = c at ; c; y(t) = t v; v 2 R n : y (t) = t v; t v = t Av; Av = v: y A v p A; p() = (A I n ): ; : : : ; n v () ; : : : ; v (n) v () ; : : : ; v (n) ' (i) (t) = i t v (i) ; i = ; : : : ; n; t 2 R;

V y y = c ' () + + c n ' (n) ; c ; : : : ; c n : ; : : : ; n A = a + b A v = u + w A v = uw y(t) = (a+b)t (u + w) y (t) = Ay(t); y y y(t) = at (bt) + (bt) (u + w) h i = at (bt)u (bt)w + (bt)u + (bt)w : y () y (2) ; y () (t) = at (bt)u (bt)w ; y (2) (t) = at (bt)u + (bt)w ; v = u w: at (bt) at (bt) A; A n ; : : : ; n ; n V;

( y (t) = Ay(t); t 2 R; B C A := @ A y () B C := @ A: y() = y () ; p A p() = ( )[( ) 2 + ]; A = ; 2;3 = : = v 2 R 3 (A I 3 )v = (A I 3 )v = B CB @ A@ v v 2 v 3 C A = ; v 2 = v 3 = v v v = ; ' () (t) = t B C @ A; t 2 R; 2;3 = 2 = + ; 3 = = 2 v 2 C 3 (A 2 I 3 )v = A ( + )I 3 v = B CB @ A@ v v 2 v 3 C A = ; v = v 2 = v 3 : v 3 := ; v 2 = ; y(t) = (+)t B C @ A; t 2 R;

" # y(t) = (+)t B C @ A = t B C B C ( t + t) @ A + @ A " # " # = t B C B C t @ A t @ A + t B C B C t @ A + t @ A ; y(t) = t B C @ ta + t B C @ ta: t t ' (2) (t) = t B C @ ta ' (3) (t) = t B C @ ta; t 2 R; t t y(t) y(t) = c t B C @ A + c 2 t B C @ ta + c 3 t B C @ ta; t 2 R; t t c ; c 2 c 3 : c ; c 2 c 3 t = c B C B C B C B C @ A + c 2 @ A + c 3 @ A = @ A B @ c c 3 c 2 C B C A = @ A;

c = c 2 = c 3 = : y(t) = t B C @ A + t B C @ ta + t B C @ ta = t B C @ t ta; t 2 R: t t t + t ; : : : ; n A A A n A A k k < n: k t v; v 2 C n : n k n v A ; (A I n ) v = : Πλήθος γραμμικά ανεξάρτητων γενικευμένων ιδιοδιανυσμάτων. ; : : : ; k A 2 R n;n ; ; : : : ; k ( + + k = n) ; : : : ; k ( j j ); j < j ; (A j I n ) 2 v = j + (A j I n ) m v = ; m < j ; m j (m j < j ) (A j I n ) m+ v = m j + j A j : n A: n

= A: A n n t v; A v t v; = A k; k < n; k t v: A; v; (A I n ) 2 v = (A I n )v : v; ta v = t v + t(a I n )v = (A I n ) 3 v = (A I n ) 2 v : v; ta v = t v + t(a I n )v + t 2 2 (A I n) 2 v = n ( y 2 3 (t) = Ay(t); t 2 R; B C A := @ 2 A y () B C := @ 2A: y() = y () ; 2

p A p; p() = (2) 3 ; = 2 A v 2 R 3 (A I 3 )v = (A 2I 3 )v = 3 v B CB C @ A@ v 2 A = ; v 2 = v 3 = v v v = ; ' () (t) = 2t B C @ A; t 2 R; A; v 2 R 3 (A 2I 3 ) 2 v = (A 2I 3 )v : (A 2I 3 ) 2 = ; (A 2I 3 ) 2 v = B CB @ A@ v 3 v v 2 v 3 C A = ; v 3 = v v 2 v = v 2 = A: ' (2) (t) = 2t v + t(a 2I 3 )v 3 = 2t B C @ A + t 2t B CB C @ A@ A = 2t B C @ A + t 2t B C @ A; t ' (2) (t) = 2t B C @ A; t 2 R;

(A 2I 3 ) 2 v = ; A v 2 R 3 (A 2I 3 ) 3 v = (A 2I 3 ) 2 v : (A 2I 3 ) 3 = ; v 2 R (A 2I 3 ) 3 v = : v = (; ; ) T (A 2I 3 ) 2 v (A 2I 3 ) 2 v = ' (3) (t) = 2t v + t(a 2I 3 )v + t 2 2 (A 2I 3) 2 v 2 3 3 = 2t 6B C B CB C 4@ A + t @ A@ A + t 2 B CB C7 @ A@ A5; t 2 R; 2 3t ' (3) (t) = 2t B t 2 2 C @ t A; t 2 R; y(t) 2 t 3t y(t) = 2t 6 B C B C B t 3 2 2 C7 4c @ A + c 2 @ A + c 3 @ t A5 ; c ; c 2 c 3 : + 5t ' (3) (t) = 2t B t 2 2 C @ 2 t A; t 2 R;

y () (t); : : : ; y (n) (t) y (t) = Ay(t); A 2 R n;n ; y y(t) = c y () (t) + + c n y (n) (t); c ; : : : ; c n ; y(t) = Y (t)c; Y (t) n n y () ; : : : ; y (n) c = (c ; : : : ; c n ) T 2 R n : Θεμελιώδης πίνακας. n n Y (t) Y (t) ta : Θεμελιώδης πίνακας και εκθετική συνάρτηση. Y (t) ta = Y (t)y () : Y (t) t: s y (t) = Ay(t); t 2 R; y(s) = v; v 2 R n ; t = s; Y (s)c = v: v 2 R n ; Y (s) s Y (t) t: Y (t) = y () (t); : : : ; y (n) (t) = Ay () (t); : : : ; Ay (n) (t) = AY (t); Y (t) = AY (t):

ta ( ta ) = A ta : Y (t) Z(t) C 2 R n;n ; Z(t) = Y (t)c: y () (t); : : : ; y (n) (t) Y (t) z () (t); : : : ; z (n) (t) Z(t) Z(t) Y (t); z (j ) (t) = c j y () (t) + + c nj y (n) (t); j = ; : : : ; n; Z(t) = Y (t)c C = (c ij ) i;j =;:::;n : ta = Y (t)c: t = ; C = Y () ; ta = Y (t)y () : y (t) = Ay(t) + f (t); A 2 R n;n ; f : R! R n : y y y + y : y : y () (t); : : : ; y (n) (t) y(t) y(t) = c y () (t) + + c n y (n) (t); t 2 R; c ; : : : ; c n c i v i y y (t) = v (t)y () (t) + + v n (t)y (n) (t);

y (t) = Y (t)v(t); Y (t) = y () (t); : : : ; y (n) (t) v(t) = v (t); : : : ; v n (t) T : Y (t)v(t) + Y (t)v (t) = AY (t)v(t) + f (t); Y (t) Y (t)v (t) = f (t); v (t) = Y (t) f (t): v Z v(t) = Y (t) f (t) dt: y Z y (t) = Y (t) Y (t) f (t) dt y; c Z B C y(t) = Y (t) @ : A + Y (t) Y (t) f (t) dt; c n c ; : : : ; c n ( y (t) = Ay(t) + f (t); t 2 R; y(t ) = y () ; y(t) = Y (t)y (t ) y () + Y (t) Z t t Y (s) f (s) ds:

t = ; Y (t)y () = ta ; Y (t)y (s) = Y (t)y () Y ()Y (s) = Y (t)y () Y (s)y () = ta sa = (ts)a ; y(t) = ta y () + Z t (ts)a f (s) ds: y (t) = Ay(t) + f (t); B C y() = @ A; B C B C A := @ 2 2A f (t) := @ A; 3 2 t (2t) t 2 R: ta : p A p() = ( )( 2 2 + 5); A = 2;3 = 2: = v 2 R 3 (A I 3 )v = (A I 3 )v = B CB @ 2 2A@ 3 2 v v 2 v 3 C A = ; v = v 3 v 2 = 3v 3 /2: v 3 = 2; 2 y () (t) = t B C @ 3A; t 2 R; 2 y (t) = Ay(t) 2;3 = 2 2 = + 2; 3 = 2 = 2

v 2 C 3 (A 2 I 3 )v = A ( + 2)I 3 v = 2 B CB @ 2 2 2A@ 3 2 2 v v 2 v 3 C A = ; v = v 3 = v 2 : v 2 := ; v 3 = ; y(t) = (+2)t B C @ A; t 2 R; y(t) = (+2)t B C @ A = t (2t) + (2t) " # B C B C @ A + @ A " # " # = t B C B C (2t) @ A (2t) @ A + t B C B C (2t) @ A + (2t) @ A ; y(t) = t B C @ (2t) A + t B C @ (2t) A: (2t) (2t) y (2) (t) = t B C @ (2t) A y (3) (t) = t B C @ (2t) A; t 2 R; (2t) (2t) y (t) = Ay(t) 2 Y (t) = t B C @ 3 (2t) (2t) A: 2 (2t) (2t)

2 Y () B C = @ 3 A 2 2 B 3 C = @ 2 A; ta = Y (t)y () = t B @ 3 + 3 C (2t) + (2t) (2t) (2t) 2 2 A: + 3 (2t) (2t) (2t) (2t) 2 y Z t y(t) = ta B C @ A + ta sa f (s) ds Z t = ta B C @ A+ ta s B @ 3 + 3 CB C (2s) (2s) (2s) (2s) 2 2 A@ Ads 3 (2s) (2s) (2s) (2s) s (2s) 2 Z t = t B C @ (2t) (2t) A + ta s B C @ (2s) (2s) A ds (2t) + (2t) 2 (2s) = t B C @ (2t) (2t) A + ta B (4t) C @ 8 A; (2t) + (2t) t + (2t) 2 8 y(t) = t B @ (2t) ( + t ) (2t) C 2 A: ( + t ) (2t) + 5 (2t) 2 4 p : [a; b]! R y (t) = p(t)y(t); t 2 [a; b]; C: R t y(t) = C a p(s) ds

y u; u(t) = R t a p(s) ds y(t); t 2 [a; b]; u = ; u Η μέθοδος της μεταβολής των σταθερών p; q : [a; b]! R y (t) = p(t)y(t) + q(t); t 2 [a; b]; C ; t y(t) = R a p(s) dsh Z t C + q(s) R i s a p() d ds ; a t b; a y(t) = C (t)r t a p(s) ds ; C C C y (t) = y(t) + [y(t)] 2 ; t 2; y() = y : [a; b] [; 2]; y c < / c > / 2 : y = p /( p ) t 3/2: c = /( p ): y = :

y (t) = y(t) [y(t)] 2 ; y() = 2; I; y: y (t) = t 2 y(t) [y(t)] 2 ; t 2; t y() = y : y = 3 t p 2: c = /4: y = 3: y (t) = + [y(t)]2 ; 2ty(t) y (t) = 2t [y(t)] 3 y(t) ; y (t) = 3[y(t)]2 + t 2 : 2ty(t) y() = ; y (t) = 2t + y(t) t + 2y(t) :

y (t) = t + [y(t)]2 ; ty(t) = (t): ( y (t) = Ay(t); t 2 R; y() = y () ; ( y (t) = Ay(t); t 2 R; y() = y () ; 2 B C A := @ 2 A y () B C := @ 2A: 2 B C A := @ 3 2A y () B C := @ 2A: I R p : I! R A nn p(t)a = p (t)a p(t)a ; t 2 I: X p(t)a [p(t)a]` X = = [p(t)]` A` `! `! `= `= p(t)a = X `= A` `p (t)[p(t)]` `! k= = p (t)a X [p(t)a]` `= X = p [p(t)a] k (t)a = p (t)a p(t)a : k! y ( y (t) = p(t)ay(t); t 2 I; y(a) = y () ; (` )! a I; y(t) = (R t a p() d)a y () ; t 2 I:

y ( y (t) = p(t)ay(t) + f (t); t 2 I; y(a) = y () ; a I f : I! R n A n n ; : : : ; k A v () ; : : : ; v (k) v () ; : : : ; v (k) v () v () ; : : : ; v (`) ; ` < k; c v () + + c`+ v (`+) = ; c Av () + + c`+ Av (`+) = ; c v () + + c`+ `+ v (`+) = : `+ c (`+ )v () + + c`(`+ `)v (`) = : c i (`+ i ) = ; i = ; : : : ; `; c = = c` = : c`+ v (`+) = ; c`+ = : v () ; : : : ; v (`+) n n M; M n: M x = (x ; ; : : : ; ) T M (; ; : : : ; ) T = M 2 x = (x ; x 2 ; : : : ; ) T ; M n x = n: x 2 C n M: Αλγεβρική και γεωμετρική πολλαπλότητα ιδιοτιμών ενός πίνακα Jordan. nn J = (J ; J ; : : : ; J k ); J J = ( z ; : : : ; z m ); J` ` = ; : : : ; k:

p() := (J I n ) J I n ; p() = ( z ) ( z m )( ) n ( k ) n k : J z ; : : : ; z m ; : : : ; k z i j J J: J J J J ; : : : ; J k ; J J J n n A; A = S JS; A Για κάθε n n μηδενοδύναμο πίνακα A ισχύει A n = : A; B; C nn A C ABC = ; B B = : ABC = A C : nn J; J J n = : x 2 R n ; y := J x y i = x i+ ; J (i; i + ) y i = ; J (i; i + ) J 2 x; : : : ; J n x J n x = ; x 2 R n ; J n = : A nn A A n = :! A = 2 2 ; A = 2 2 5 3 2 B C @ 5 9 6A; 6 4

A; S AS = J: A J S A n S = J n : J n = ; A n = : A n n A A x A`x = `x: A` ; ` 2 N: Γενικευμένα ιδιοδιανύσματα ενός πίνακα Jordan. n n J = (J ; J ; : : : ; J k ); J J = ( z ; : : : ; z m ); J` ` = ; : : : ; k: n J n` J`; ` = ; : : : ; k: J; J J J` ; (J I n ) + : J` + J` I n` (J` I n`) + = : J I n ; J : (J I n ) + J : (J I n ) + (J I n ) + v = ; J J ; n n J C n ; J R n ; J Πλήθος γενικευμένων ιδιοδιανυσμάτων οποιουδήποτε n n πίνακα. S n n v () ; : : : ; v (k) 2 C n Sv () ; : : : ; Sv (k)

A nn n A; S AS = J A: A J S (AI n )S = J I n ; S (AI n )`S = (J I n )`; (A I n )`S = S(J I n )`: v () ; : : : ; v (n) 2 C n J; Sv () ; : : : ; Sv (n) 2 C n A;