Differential Equations
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Transcript of Differential Equations
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1.
(separable equations
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)
2. (1)
. (2)
(reducible to separable form). :
.
t=T+h y=Y+k a1b2a2b1, z=a1t+b1y a1b2=a2b1.
3.
1 (1) exact ()
.. .
u .
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(1) exact. t,
(2). (2) y, :
(3). (2)
k(y) (3) y.
(exact differential equations)
4. exact F(t,y)0 ( ).
,
(1).
t y. F=F(t) (1)
. F=F(y).
(integrated factors)
5. 1 , .
r(t)=0 , .
(linear first-order differential equations)
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:
. :
, F(t) exact.
. :
. .
6. (1). y1(t) y2(t)
(1) (a, b) ( ) w[y1, y2]0 , : y(t)=c1y1(t)+c2y2(t). y1(t) y2(t), f(t) g(t) (. ). () (1) y1(t)
y2(t)=v(t)y1(t) ( ).
2 (linear second-order differential equations)
y1(t) y2(t) (t) . , . (t) (t)=v1(t)y1(t)+v2(t)y2(t)
v1 v2, .
v1 v2. .
: v1(t) v2(t)
( - method of variation of parameters).
7.
1 2. y(t)=y1(t)+y2(t) :
2
) 12, . ) 1=2=, . ) =j, .
. (t), ( ):
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) r(t)=0+1t+ .. +nt
n. : c0 (t)=A0+A1t+ +Ant
n. c=0 b0 (t)=t(A0+A1t+ +Ant
n). c=0 b=0 (t)=t2(A0+A1t+ +Ant
n). ) r(t)=(0+1t+ .. +nt
n)et. , y(t)=etv(t) : a2+b+c 0 (t)=(A0+A1t+ +Ant
n)et. a2+b+c=0 2a+b0 (t)=t(A0+A1t+ +Ant
n)et. a2+b+c=0 2a+b=0 (t)=t2(A0+A1t+ +Ant
n)et.
) r(t)=(0+1t+ .. +ntn)sin(t). (t)
(t) r(t)=(0+1t+ .. +nt
n)ejt.
) r(t)=(0+1t+ .. +ntn)cos(wt). (t)
(t) r(t)=(0+1t+ .. +nt
n)ejwt.
y(t)=y1(t)+y2(t)+(t).
8.
(1) ai . y(t)=et . y(t)=et (1), , . () 1, 2, .. k ( ), .. n, (1)
, . . ., ckt1ekt+ +ck+1ek+1t+cnent. k=+j k=-j . k k , (1) :
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n
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