Q2ts - Quiz II 1. a) The differential form = (ln y + y x-...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

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

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Quiz II 1. a) The differential form = (ln y + y x- 1) dx + (ln x + x y- 1) dy is exact in any rectangle contained in the first quadrant (where x > 0 and y > 0). Find a function F ( x,y ) defined in the first quadrant such that dF = . Solution . Following the standard procedure F ( x,y ) = Z (ln y + y/x- 1) dx + h ( y ) = x ln y + y ln x- x + h ( y ) F y = x/y + ln x + h ( y ) So we need h ( y ) =- 1, which gives h ( y ) =- y + C and F ( x,y ) = x ln y + y ln x- x- y + C. The level curves of the function F ( x,y ) that you found in part a) are solutions to a first order differential equation. Write down that differential equation. Solution . We have = 0 when y = y ( x ) is a solution to the differential equation. So 0 = (ln y + y/x- 1) dx +(ln x + x/y- 1) dy dx dx = [(ln y + y/x- 1)+(ln x + x/y- 1) dy dx ] dx So the differential equation is 0 = (ln y + y/x- 1) + (ln x + x/y- 1) dy dx or in normal form dy dx =- ln y + y/x- 1 ln x + x/y- 1 ....
View Full Document

This note was uploaded on 05/01/2008 for the course MATH 33B taught by Professor Staff during the Spring '07 term at UCLA.

Page1 / 2

Q2ts - Quiz II 1. a) The differential form = (ln y + y x-...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online