1.

(separable equations

.

)

2. (1)

. (2)

(reducible to separable form). :

.

t=T+h y=Y+k a1b2a2b1, z=a1t+b1y a1b2=a2b1.

3.

1 (1) exact ()

.. .

u .

.

(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)

:

. :

, 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), ( ):

) 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) :

.

n

.