p2 - Math 5525: Spring 2010. Introduction to Ordinary...

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Unformatted text preview: Math 5525: Spring 2010. Introduction to Ordinary Differential Equations: Homework #2 (due on March 10). 100 points are divided between 8 problems. #1. (10 points). Find the solution of the problem xy + 4y + x = 0. satisfying the initial condition y(1) = 0.: #2. (10 points). Find the general solution of the equation dy x + y2 = . dx 2xy #3. (10 points). Find the general solution of the equation y + 4y + 5y = e-2x cos x. #4. (10 points). Find the solution of the initial value problem 2 , y(0) = y (0) = 0. cos3 x #5. (10 points). Let y1 (x) and y2 (x) be smooth functions on R1 , such that y +y = W = y1 y2 y1 y2 0, y1 (x) = 0 for all x R1 . Show that y2 Cy1 , where C = const. #6. (10 points). Show that the solution y(x) of the initial value problem y = x2 + y 2 , cannot be extended to [0, +). #7. (20 points). Consider the initial value problem for the integral equation x y(0) = 0 y(x) + 0 y(t)dt = y 2 (x), y(0) = y0 . (a) (3 points). Find all possible values of y0 for which this problem has a solution. (b) (10 points). By differentiating both sides of the integral equation, derive a differential equation. Find a general solution of this equation. (c) (7 points). For a nontrivial (not identically zero) solution y(x), show that y(x) 1 + x for all x 0, and find the limit of y(x)/x as x . #8. Let f (x) be a continuous function on [0, +), such that |f (x)| M = const for all x. (a) (10 points). Let y(x) be a solution of the equation y + y = f on [0, +). Show that |y(x)| max{|y(0)|, M } for all x 0. (b) (10 points). Let y(x) be a solution of the equation y + 2y + y = f on [0, +). Show that y(x) is bounded on [0, +). ...
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