HW AnswerGuide1 - Answer Guide 1 Math 427K Unique Number 55160 Wednesday 1 Suppose that the area bounded by a smooth curve y(x the x-axis the

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

View Full Document Right Arrow Icon
Answer Guide 1 Math 427K: Unique Number 55160 Wednesday, August 31, 2011 1. Suppose that the area bounded by a smooth curve y ( x ), the x -axis, the y -axis, and the vertical line through x equals twice the curve’s arc length between 0 and x .F i n dt h e diferential equation satis±ed by the curve. Solution: Writing ° x 0 y x )dˆ x =2 ° x 0 ± 1+( y ° x )) 2 x and applying the ²undamental theorem o² calculus, one obtains y ( x )=2 ± 1+( y ° ( x )) 2 . Simpli²ying, we get ( y ° ( x )) 2 = y ( x ) 2 4 1 . ( ° ) 2. A curve in the x, y -plane is de±ned by the condition that the sum o² the x - and y -intercepts o² its tangents always equals m . Find a diferential equation ²or the curve. Solution: Note that a correct answer here may be written in more than one ²orm. The tangent line so a smooth curve y ( x ) at an arbitrary point ( a, b ) has the ²orm y b = y ° ( a )( x a ) . The y -intercept, ²ound by setting x = 0, is y int ( a, b )= b ay ° ( a ) . The x -intercept, ²ound by setting y = 0, is x int ( a, b )= a b y ° ( a ) . (You may assume that y ° ( a ) is never zero, since otherwise the x -intercept does not exist.) Note that we have ²ound y int and x int as ²unctions o² the point ( a, b ). Since ( a, b ) was arbitrary, we can substitute x = a , y = b , and dy dx = y ° ( x ) to get a nicer ²ormula m = x ² 1 dy dx ³ + y ² 1 dx
Background image of page 1

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

View Full DocumentRight Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 10/24/2011 for the course MATH 427K taught by Professor Delallave during the Spring '11 term at University of Texas at Austin.

Page1 / 4

HW AnswerGuide1 - Answer Guide 1 Math 427K Unique Number 55160 Wednesday 1 Suppose that the area bounded by a smooth curve y(x the x-axis the

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