BO MON TOAN NG DUNG - HBK-------------------------------------------------------------------------------------

TOAN 4 HK2 0506

CHUOI VA PHNG TRNH VI PHAN BAI 4: PHNG TRNH VI PHAN CAP 1

(PHAN 1)

TS. NGUYEN QUOC LAN (4/2006)

NOI DUNG---------------------------------------------------------------------------------------------------------------------------

1 TONG QUAN

2 CAC KHAI NIEM C BAN

3 NGHIEM PHNG TRNH VI PHAN

4 PHNG TRNH VI PHAN CAP 1

5 PHNG TRNH VI PHAN PHAN LY BIEN SO

6 PHNG TRNH VI PHAN CAP 1 TUYEN TNH

7 PHNG TRNH VI PHAN TOAN PHAN

8 PT VI PHAN KHONG GIAI C VI AO HAM

(SINH VIEN T XEM: SGK, TRANG 136 139)

TONG QUAN -----------------------------------------------------------------------------------------------------------------

Mo hnh Vat Ly, C, ien Phng trnh vi phan!

Van toc nguoi i ty le thuan vi hieu nhiet o vat va

nhiet o khong kh. Biet nhiet o khong kh la 20C va

vat giam nhiet o t 100C xuong 60C sau 20 phut.

Sau bao lau t thi iem au, nhiet o vat se la 30C?

R

L0V

C

iC

qU

dt

diLUiRU CLR === ,,

Kirchhoff: 0=++C

qRi

dt

diL

022

=++=C

q

dt

dqR

dt

qdL

dt

dqi

Phng trnh vi phan (thng ODE): ham an y = y(x),

bien x & cac ao ham (hoac vi phan) y(k), k = 0, 1 n

VD: 03 =+ xdx

dy ( ) xexyyy =++ 3'4'' ( ) ( ) 0=+ dyyxdxyx

KHAI NIEM C BAN --------------------------------------------------------------------------------------------------------------------------

Cap 1 Cap 2 Cap 1

Phng trnh vi phan cap n: cha ao ham cao nhat cap n

Dang tong quat PT vi phan cap 1:

( ) ( ) ( ) ( )( )( ) 0,,'',',, =xyxyxyxyxF nKDang tong quat cap n:( ) ( )( ) 0',, =xyxyxF

NGHIEM PHNG TRNH VI PHAN -------------------------------------------------------------------------------------------------------------------------

Nghiem PTVP:

Ham so y = y(x),

x khoang I R (c) Dang tham so

(a) Dang hien: y = f(x)

(b) Dang an: H(x, y) = 0( )( )

==

tyy

txx

VD: xeydx

dy =Nghiem: xey 2=

Nghiem: xx eCey 2+= nghiem tong quat

VD: xdx + ydy = 0: 2 dang nghiem hien, an VD: 21' yy =

Nghiem PTVP cap n THONG THNG cha n hang so:

o th nghiem: ng cong tch phan( ).,,, 1 nCCxy K=

PHNG TRNH PHAN LY BIEN SO--------------------------------------------------------------------------------------------------------------------

Phng phap: Phan ly x & dx mot ve, y & dy mot ve.

Tch phan 2 ve Nghiem (noi chung dang an)

VD: Kiem tra dang phan ly cua cac ptrnh xyya ='/

( ) ( ) 011/ 22 =++ dyxydxyxb ( ) 04/ =++ dxxyxdyc

( ) ( )( ) ( )( ) ( ) ( ) ( )

=+=+

=

0

0

'

2211 dyygxfdxygxf

dyygdxxf

ygxfyTong quat: 3 dang

phng trnh vi phan

phan ly bien so

Nhan dang: Bien x va y phan ly (separable)

Co the tach ri moi ve 1 bien! VD: 02 = dxyxdy

GIAI PT VI PHAN PHAN LY BIEN SO-----------------------------------------------------------------------------------------------------------------

VD (SGK, 23/tr190): Van toc nguoi i cua vat ty le

thuan vi hieu nhiet o cua vat va nhiet o khong kh.

Biet nhiet o khong kh la 20C va vat giam nhiet o

t 100C xuong 60C sau 20 phut. Hoi sau bao lau ke

t thi iem au, nhiet o cua vat se la 30C?

VD: xya 3sin'/ = yeyb ='/x

yyc

2'/ =

VD: ( ) 05cos2/ 4 =++ dyydxxxa

( ) ( ) 0/ 2222 =++ dyyxxdxxyyb xyxyyc 2'/ 2 =

OI BIEN A VE PHAN LY -------------------------------------------------------------------------------------------------------------------------

VD: (x2 + y2)dx xydy = 0: Chu y P(x, y) = (x2 + y2), Q = xy!

Cha tong: y = f(ax + by + c) oi bien: u = ax + by + c

VD: y = (2x + 3y + 1)2 2(2x + 3y + 1)

Ty so: oi bien:

=x

yfy' uxuyuxy

x

yu +=== ''

ac biet: P(x, y), Q(x, y) tong xy, + = n Phng

trnh ang cap Pdx + Qdy = 0: Dang y = f(y/x)!

VD:xy

xyyyb

2'/

2 +=x

yya += 1'/

PT VI PHAN CAP 1 TUYEN TNH -------------------------------------------------------------------------------------------------------------------------

y = a(x)y + b(x) (E): khong thuan nhat (co ve phai)

PT thuan nhat (khong ve phai) tng ng: y = a(x)y (E0)

Nhan dang: y = f(x, y): Ve phai ch cha y bac 1 ( t so)

y = f(x, y) = a(x)y + b(x): tuyen tnh (bac 1) theo y

Tuyen tnh theo x = x(y)!

VD: Xac nh phng trnh tuyen tnh:

xexyyc =+3'/

32'/ xyx

ya =

( ) 022/ 2 =+ dyxyydxd32'/ xyeyb x =+

NGHIEM TONG QUAT THUAN NHAT ------------------------------------------------------------------------------------------------------------------

VD: Giai cac PTVP thuan nhat:x

yya ='/ xyyb tg'/ =

PT cap 1 tuyen tnh thuan nhat: y + a(x)y = 0 (E0)

co nghiem tong quat dang: ( ) sohang:,0 CxCyy =

VD: T nghiem tong quat cac PT thuan nhat tren, tm 1

nghiem rieng (nghiem ac biet) cua PT khong thuan nhat33'/ x

x

yya +=

x

exyyb

x

costg'/ +=

N0 rieng yr = C(x)y0(x): bien thien hang so ytq.tn = Cy0(x)

Thay yr = C(x)y0(x) vao (*) ( ) ( ) ( )xbxyxC =0'

TONG KET --------------------------------------------------------------------------------------------------------------------------------

Cong thc nghiem tong quat PTVP cap 1 tuyen tnh:

( )

+=+=

dxxpdxxpdxxpedxexfCeyxqyxpy

)()()()()('

PTVP cap 1 t/tnh (E):

2/ Bien thien hang so C = C(x)

1/ PT thuan nhat:

3/ Nghiem sau cung: Tong 2 nghiem bc 1 & bc 2

( ) ( ) ( ) ( )xqyxpyxbyxay =++= '' hoac( ) ( ) ( )xCyyyxpyyxay 00'' ==+= hoac

( ) ( ) == KxCxbyC 0'

Nghiem tong quat PT tuyen tnh = Nghiem tong quat PT

thuan nhat (de)+ Nghiem rieng PT khong thuan nhat (kho)

V DU---------------------------------------------------------------------------------------------------------------------

VD: Giai ( ) ( )412'1 +=+ xyyx

1/ Phng trnh thuan nhat: ( )xCyyyx

y 0012

' ==+

2/ Bien thien hang so: ( ) ( ) ( ) ( )xCxyxCxCC +== 30 1'

3/ Nghiem sau cung: ( ) ( )42 121

1 +++= xxCy

VD: Giai cac phng trnhx

xy

xdx

dya

sin1/ =+

( ) yyxyb =+ 2'/

VD: Tnh y(2) vi ham y thoa: ( ) 11,3' ==+ yxx

yy

( )311

2' +=

+ xy

xy