k
a0 + a1x + a2x2 ++ anxn + , :
a0 + a1 + a22 ++ ann + = n=0S¥ ann.
l - , , = l
, a0 + a1l + a2l2 ++ anln + = n=0S¥ anln (*) l .
: l - . = n=0S¥ anln, . . . n nx = lnx, = n=0S¥ anln Þ (*).
, Rk. a0 + a1 + a22 ++ ann + = n=0S¥ ann , Rk a0 + a1 + a22 ++ ann +
,
a0 + a1 + a22 ++ ann + (**) . . (**) , . , (**) Þ .
.
, . .
( ).
(t) , . , , .. , , , . (t), I(t) (.. ) . I(t)= 0 , . I<0 . , I.
, .. .
y¢=mI (m=const), 1/m . a - , .. , , I=a py.
I, y¢=a m y¢=, =ma =const.
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Dy/y=kdt. ln|y|=kt+lnC, y=Cekt.
y(t0)=y0, C=y0e-kto, .. y=y0ek(t-to) . , .
.
=() (dp/dy <0), .. . :
y¢=kp(y)y,( k=la.) . k>0, p>0, y>0, (t) (y¢>0). (t) . t,
y¢¢=ky¢(dp y +p) y¢¢=ky¢p(dp * y +1), .. y¢¢=ky¢p(1- 1),
dy dy p |ey|
ey(p)= dy * p - .
dp y
, , .. |ey|>1, y¢¢>0, .. . , .. |ey|<1, y¢¢<0 .
, , ()=b-ay (a, b>0), :
y¢=k(b-ay)y. , y¢=0, =0 = b/a, , ¢¢<0 = b/2a, ¢¢>0 > b/2a. y(t). ,
dy = kdt, dy (1 + a)= kdt.
y(b-ay) b b-ay
,
Ln|y|-ln|b-ay|= kbt+lnC, .. y/(b-ay)=Cekbt. y= Cekbt.
1+Caekbt
. , , ..
, t , t , =b/a. y¢=k(b-ay)y ()=0. > b/a, . ()<0, .
, , . ()= a+b - (b,a - )
¢=k(p(y)y-a-b). p(y)=b-a, 2. .
1) D<0. , ¢<0. , .
2) D=0. ¢ < 0 =* < b/a. , (t0)=y0>y*, * +, , (t0)< * * -.
3) D>0. =1, =2. (0<y1<y2). ¢>0 y1<<y2 ¢<0 <y1 >2.
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3. .
Y=F(K,L) , (), L , F(K,L) - (F(tK,tL)=tF(K,L)). f(k) :
F(k)= F(K,L)/L=F(K/L,1)=F(k,1), k=K/L . , f¢(k)>0, f¢¢(k)<0.
, :
1. , .. L¢=aL(a=const);
2. , , .. L¢=K¢+bK (b - ).
l (.. I=lY), lY=K¢+bK ÞK¢=lY-bK.
ln k=lnK-lnL.
t, k¢/k=K¢/K-L¢L. L¢ K¢, k¢ = lY-bK - a, .. k¢= lYk (b+a)k = lYK -(b+a)k
k K K kL
, f=Y/L, K¢=lf(k)- (b+a)k. .