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# HW2 - a y = 2 xy y(0 = 0 y(0 = 2 y(0 = 1 y(1 =-5 b y y x =...

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Assignment 2 Math 427K: Unique Number 58780 Due Monday, February 4, 2008 To obtain full credit, you must show all your work. Your assignments will not be accepted unless they are legible, neat, and stapled. Problem 1. Use separation of variables to find the general solution. Then, obtain the particular solution satisfying the given initial condition. Sketch the graph of the solution. a) y 0 - 3 x 2 e - y = 0 , y (0) = 0 b) y 0 = ( y 2 - y ) e x , y (0) = 2 c) y 0 = e x +2 y , y (0) = 1 d) y 0 = y ( y - 2) , y (0) = 4 Problem 2. Predict existence and uniqueness of a solution to the ODE through each given point. Does there exist a solution curve through the first initial point? Is it unique? Through the second point? And so on. Further, find the general solution of the ODE and the particular solution, if any, though the given initial points. Give the internal of existence of those particular solutions. Provide either a labeled sketch or a computer plot of those solutions or the given ODE in the specified rectangular region in the x, y plane as well as the integral curve through the specified point P

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Unformatted text preview: . a) y = 2 xy y (0) = 0 , y (0) = 2 , y (0) = 1 , y (1) =-5 b) y y + x = 0 y (0) = 4 , y (3) = 0 , y (-3) = 0 , y (-3) =-4 c) y = tany y (3) = 2 , y (3) = 0 , y (0) = 2 . Problem 3. Show that equation is exact and obtain its general solution.Also, ﬁnd the particular solution corresponding to the given initial condition. a) ( sin ( xy ) + xy cos ( xy )) dx + x 2 cos ( xy ) dy = 0 , y (2) =-1 b) 3 x 2 sin (2 y )-2 xy ) dx + (2 x 3 cos (2 y )-x 2 ) dy = 0 , y (1 / 2) = 3 . 1 c) 3 x 2 y ln ( y ) dx + ( x 3 ln ( y ) + x 3-2 y ) dy = 0 , y (8) = 3 d) 3 dx-dy = 0 , y (0) = 6 1 2 Problem 4. (Euler’s Method.) Use the Euler method to compute, by hand, y 1 , y 2 and y 3 for the speciﬁed initial value problem using h = 0 . 2. a) y =-y, y (0) = 1 b) y = 1 + 2 xy 2 , y (1) =-2 c) y = x 2-y 2 , y (3) = 5 d) y = tan ( x + y ) , y (1) = 2 e) y = √ x + y, y (0) = 3 Problem 5. Solve problems 1 to 7 from Chapter 2, Section 5 of the course textbook....
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