ch2_6 - 2 . 6 : E x a c t E q u a t i o n s F o r m : d y d...

This preview shows pages 1–3. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 2 . 6 : E x a c t E q u a t i o n s F o r m : d y d x = P ( x , y ) Q ( x , y ) o r P ( x , y ) + Q ( x , y ) d y d x = ( 1 ) D i ff e r e n t i a l f o r m f o r m u l a t i o n : P ( x , y ) d x + Q ( x , y ) d y = ( 2 ) L e v e l c u r v e s o f a f u n c t i o n F : F ( x , y ) = C ( 3 ) O n c u r v e s ( 3 ) : d F F x d x + F y d y = ( 4 ) C u r v e s ( 3 ) a r e i n t e g r a l c u r v e s f o r ( 1 ) i f ( 3 ) d e fi n e s i m p l i c i t s o l u t i o n s o f ( 1 ) . ( 2 ) i s e x a c t i f e x i s t s F s . t . P = F / x , Q = F / y I f ( 2 ) i s e x a c t , t h e n ( 3 ) d e fi n e s i n t e g r a l c u r v e s C o n d i t i o n f o r e x a c t n e s s : P y = 2 F x y , Q x = 2 F x y P y = Q x ( 5 ) 1 F i n d F ( i f ( 5 ) i s s a t i s fi e d ) : F / x = P F ( x , y ) = H ( x , y ) + ( y ) H ( x , y ) = integraltext P ( x , y ) d x F / y = Q y H ( x , y ) + ( y ) = Q ( x , y ) ( y ) = Q ( x , y ) y H ( x , y ) S o l v e t h i s f o r ( y ) ( r . h . s . d e p e n d s o n l y o n y ) E x . : e y + ( x e y s i n y ) d y d x = o r e y d x + ( x e y s i n y ) d y = H e r e : P ( x , y ) = e y Q ( x , y ) = x e y s i n y C h e c k e x a c t n e s s : P / y = e y , Q / x = e y E q u a t i o n i s e x a c t F i n d F H ( x , y ) = integraldisplay e y d x = x e y ( y ) = Q H / y = ( x e y s i n y ) x e y = s i n y ( y ) = c o s y F ( x , y ) = H ( x , y ) + ( y ) = x e y + c o s y I...
View Full Document

This note was uploaded on 09/23/2011 for the course PGE 310 taught by Professor Klaus during the Spring '06 term at University of Texas at Austin.

Page1 / 9

ch2_6 - 2 . 6 : E x a c t E q u a t i o n s F o r m : d y d...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online