hw1 - y = 2 + (2 t-y ) 3 on 0 ≤ t ≤ 3 , | y | ≤ 4 P =...

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Assignment 1 Math 427K: Unique Number 58780 Due Wednesday, January 23, 2008 To obtain full credit, you must show all your work. Your assignments will not be accepted unless they are legible, neat, and stapled. Important note: DFIELD is a mathematical software copyrighted by John Polk- ing, a math professor at Rice University. It is free for educational use and may be found on-line at http://math.rice.edu/ dfield/dfpp.html Problem 1. A curve in the x,y - plane is defined by the condition that the sum of the x and y intercepts of its tangents always equals m . Find a differential equation for the curve. Problem 2. Verify that the given function y is a solution of the ODE for any value of C . Then, solve for C so that y satisfies the given initial data: a) y + y 0 = 1 , y ( t ) = 1 + C e - t , y (0) = 3 b) y y 0 = t, y ( t ) = t 2 + C, y (1) = 10 c) y 0 + 6 y = 0 , y ( t ) = C e - 6 t , y (0) = 5 Problem 3. (Direction fields.) Draw the direction fields for the given ODE in the specified rectangular region in the t,y plane as well as the integral curve through the specified point P . a)
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Unformatted text preview: y = 2 + (2 t-y ) 3 on 0 ≤ t ≤ 3 , | y | ≤ 4 P = (2 , 1) b) y + 2 y = e-t on | t | ≤ 2 , | y | ≤ 3 P = (0 , 0) c) y = tsiny on 0 ≤ t ≤ 10 , ≤ y ≤ 10 P = (2 , 2) Problem 4. Classify each equation as linear or nonlinear. a) y + e t y = 4 b) e t y = t-2 y c) y y 000 + 4 y = 3 t d) y 00 = t 3 y e) y y = t + y f) y-e y = sint Problem 5. (Bernoulli equation.) y + p ( t ) y = q ( t ) y n , where n is a constant 1 2 is called Bernoulli’s equation. a) Give the general solution of the Bernoulli’s equation for n = 0 and n = 1. b) If n is neither 0 nor 1, then the Bernoulli’s equation is nonlinear. Nevertheless, show that by transforming the dependent variable y ( t ) to v ( t ) = y 1-n , ( n 6 = 0 , 1) , the Bernoulli’s equation can be converted to the equation: v + (1-n ) p ( t ) v = (1-n ) q ( t ) which is linear. Find the general solution to the Bernoulli’s equation....
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This note was uploaded on 05/25/2008 for the course M 427K taught by Professor Fonken during the Spring '08 term at University of Texas at Austin.

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hw1 - y = 2 + (2 t-y ) 3 on 0 ≤ t ≤ 3 , | y | ≤ 4 P =...

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