[PDF] Equações Diferenciais Ordinárias

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????? ?????33? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ??

F(t;x;dxdt

;;dnxdt n) = 0 P

0(x)dnydx

n+P1(x)dn1ydx n1++Pn1(x)dydx +Pn(x)y=G(x):????? d nydx n+p1(x)dn1ydx n1++pn1(x)dydx +pn(x)y=g(x)?????

L[y]: =dnydx

n+p1(x)dn1ydx n1++pn1(x)dydx +pn(x)y

L[y] =g(x)?????

y(x0) =y0; y0(x0) =y1;;(n1)y(x0) =yn1;????? d nydx n+p1(x)dn1ydx n1++pn1(x)dydx +pn(x)y= 0????? y=c1y1(x) +c2y2(x) ++cnyn(x)????? (1:6)? ???c

1y1(x0) ++cnyn(x0) =y0

c 1y0

1(x0) ++cny0

n(x0) =y1??? c

1(n1)y1(x0) ++cn(n1)yn(x0) =yn1?????

y

W(y1;;yn) =

y 1y2yn y (n1)y1(n1)y2(n1)yn y x

0=f(t;x);?????

????f: !RN? ???? ??????? ?? ???? ?????(t;x)?? ?? ?????? RRN= R N+1? x0=f(t;x) x(t0) =x0:????? ???(t;(t))2 ? ???? ????t2I? ????0(t) =f(t;(t))? ???? ????x2 ?????(t0) =x0? Rn x(t) x' ( )t', x' ( )t, x?( )t', x' ?( )t, x????? ??? ? ??????f????? ?? (t;x)? ????? l(t;x) :x=f(t;x)(t) _x(t) =p(t)x+q(t);????? _x(t) =p(t)x+q(t) x(t0) =x0:????? (t) =eRt t0p(s)ds: e Rt t0p(s)ds(_xp(t)x) =q(t)eRt t0p(s)ds: (xeRt t0p(s)ds) =q(t)eRt t0p(s)ds: xe Rt t0p(s)dsx0=Z t t

0q(s)eRs

t0p(y)dyds: x(t) =x0eR t t0p(s)ds+Z t t

0q(s)eR

t sp(y)dyds:????? ???????q(t)0?? ????? ? ??????? ?? ????? ? ???? ???? x(t) =x0eR t t0p(s)ds; x _x=f(t)g(x);g(x)6= 0;????? g(x)dx=f(t)dt x2(;)? _x=tx ??x1(t) =pt

21?t >1

??x2(t) =pt

21?t <1

??x3(t) =pt

21?t <1

??x4(t) =pt

21?t <1?C=0

-2 -2-3 -3x t C=1 C=-1 2233
-1 -111????? ??? ????? ???????(a;b) = (1;+1)? ?? ????? ?? ???(;) = (1;1);(1;+1)? _x=f(t)g(x); g(x)dx=f(t)dt; Zx(t) x

0g(x)dx=Z

t t

0f(t)dt;

????t02(;)?x(t0) =x02(c;d)?

N(t;x)x0+M(t;x) = 0;?????

????M;N: R2

M(t;x)dt+N(t;x)dx= 0:?????

dF=M(t;x)dt+N(t;x)dx:

M(x;y)dx+N(x;y)dy

@M@y =@N@x f??? ???

M(x;y)dx+N(x;y)dy=@f@x

dx+@f@y dy ???? ????(x;y)??R? ?????

M(x;y) =@f@x

; N(x;y) =@f@y @M@y =@@y @f@x =@2f@y@x =@@x @f@y =@N@x F t=M(t;x)e Fx=N(t;x):

F(t;x) =Z

M(t;x)dt+g(x):?????

x?@F@x =@@x Z

M(t;x)dt+g0(x)

g

0(x) =N(t;x)@@x

Z

M(t;x)dt??????

F(t;x) =C:

??t? ???? @@t

N(t;x)@@x

Z

M(t;x)dt

=@N@t @@x @@t Z

M(t;x)dt

@N@t @M@x ?? ? ???????tdt+xdx= 0? ?????? ?? ? ???????xdttdx= 0??? ? ?????? (e2yycosxy)dx+ (2xe2yxcosxy+ 2y)dy= 0: @M@y = 2e2y+xysenxycosxy=@N@x ???? ?????? ??? ??????f(x;y)??? ???

M(x;y) =@f@x

?N(x;y) =@f@y @f@y =N(x;y)? ???? ? @f@y = 2xe2yxcosxy+ 2y f(x;y) = 2xZ e

2ydyxZ

cosxydy+ 2Z ydy: f(x;y) =xe2ysenxy+y2+h(x) @f@x =e2yycosxy+h0(x) =e2yycosxy; h xe

2ysenxy+y2+c= 0:

(cosxsenxxy2)dx+y(1x2)dy= 0; Y(0) = 2:

M(t;x)dt+N(t;x)dx= 0:

??t?x?? ???? ??? ? ??????? (Mdt+Ndx) = 0?????? (M)x= (N)t: M xNt+ (MxNt)= 0: (M)x=Mx;(N)t=Nt+Nddx ddt =MxNtN (x+y)dx+xlnxdy= 0;??????(x;y) =1x ??(0;1): = 1;? @N@x (x;y) =1x 1 +yx dx+ lnxdy= 0:

M(x;y) = 1 +yx

; N(x;y) = lnx; @M@y =1x =@N@x @f@x = 1 +yx =M(x;y) f(x;y) =x+ylnx+h(y) @f@y = 0 +lnx+h0(y) = lnx h

0(y) = 0?g(y) =c

x+ylnx+c= 0 ? ? ??????? ???? ? ??????? ??(0;1): x

0+p(t)x=q(t)xn;??????

x nx0+p(t)x1n=q(t)?????? (1n)xnx0+ (1n)p(t)x1n= (1n)q(t)?????? z

0= (1n)xnx0

z

0+ (1n)p(t)z= (1n)q(t)

z

0+'(t)z= (t);

????'(t) = (1n)p(t)? (t) = (1n)q(t)? z(t) =eRt

0p(u)du(Z

q(t)eR t

0p(u)dudt+K)

x

0=p(t) +q(t)x+r(x)x2??????

??? ???? ??????z=z(t)??????? ??? z=1xxp: z

0+ [q(t) + 2xpr(t)]z=r(t)

x=xp+1z _x=f(x):?????? x(t0) =x0:?????? ??????? ?????? ???x(t)? ??????? ?? ?????? ??? ? ??????? ???y(t) =x(t+c)? ??????? ??_x=f(x) x(0) =x0:?????? ????t0= 0? jx(t)xj< ?8t0? ???? ???? ? ??????? ?? ?????? ??????? ??9 >0?jx0xj< )limt!1x(t) = x? f ddt (x(t)x)2= 2(x(t)x)_x= 2(x(t)x)f(x(t))

2(x(t)x)[f(x(t))f(x)] = 2(x(t)x)f0((t));

f

0??????? >0? >0???? ???f0(x)< <0????jxxj< ?

ddt (t) (t)????tt0: ????(t)cet? ? ??? ??????? ???x(t)????? ?x??????t! 1? ?????? f _p= (abp)p;?????? p=1y dpdt =apbp2)dpp

2dt=ap

b)y2dyy

2dt=ayb;

1a lnjaybj=t+C; jpj=ejabpj+at+aC) jpj=jabpjeateaC: ???p(t0) =p0? jp0j=jabp0jeat0eaC: ??p06= 0?p06=ab jpjjp0j=jabpjjabp0jett0:?????? ????? ?????? ?? ?????p= 0?p=ab ?????? ????p06= 0?p06=ab pp

0=abpabp0ett0)p(abp0) = (abp)p0ea(tt0);

p(t) =ap0bp

0+ (abp0)ea(tt0):

? ?????? ???? ??????? ?? ????? ???p= 0? ?? ??p0>ab ab ?? ??0< p0T(0) =T0??????

T(t) = (T0Ta)ekt+Ta:??????

T ?????? ??10C? ????? ??1g?? ?????? mc(T0T) =maca(TaTa;0);?????? T dTdt +k(1 +A)T=k(Ta;0+AT0);?????? ????A=mcm aca? (dTdt +k(1 +A)T=k(Ta;0+AT0)

T(0) =T0:??????

T(t) =Ta;0+AT01 +A+T0Ta;01 +Aek(1+A)t:??????

f(T) =k(Ta;0+AT0)k(1 +A)T:

N(t;x)x0+M(t;x) = 0??????

V(t;x) =C;??????

f(t;x;) = 0;?????? ????f: ? ?? ?????? ?? ?????(t;x)? f t(t;x;) = 0?????? f (t;x;)6= 0?????? f(x;y;)y2x2= 0;x;y;2R;??????quotesdbs_dbs26.pdfusesText_32

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