ps1 - u = y/x to solve the following homogeneous equations...

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Problem Set 1 January 11, 2002 Due January 18, 2002 ACM 95b/100b 3pm in Firestone 303 Niles A. Pierce Include grading section number 1. Find the general solution and plot some integral curves for each of the following ODEs. What is the behavior of the solution as t → ∞ ? a) ty 0 - y = t 2 b) y 0 + y cos t = sin t cos t 2. Solve the following linear initial value problems and in each case describe the interval on which the solution is defined. a) y 0 + 2 xy = e - x 2 , y (0) = - 1 b) ty 0 + 2 y = t 2 - t + 1, y (1) = 1 / 2 3. Solve the following nonlinear initial value problems and in each case determine the interval on which the solution is defined. a) y 0 = 3 x 2 3 y 2 - 4 , y (1) = 0 b) y 0 = - 2 xy x 2 + 3 y 2 , y (0) = 1 c) y 0 = y + x y - x , y (2) = 2 d) y 0 = y 2 / 3 , y (1) = 8 4. The ODE y 0 = f ( x,y ) is termed “homogeneous” if f ( x,y ) can be expressed in terms of the ratio y/x . Use the substitution
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Unformatted text preview: u = y/x to solve the following homogeneous equations. a) y = 3 y 2-x 2 2 xy b) y = 4 y-3 x 2 x-y 5. The Bernoulli equation y + p ( x ) y = q ( x ) y n is an example of a nonlinear ODE that can be made linear by a change of dependent variable. a) Solve the equation for n = 1. b) Show that if n is an integer larger than one, the substitution u = y 1-n reduces Bernoulli’s equation to a linear equation. c) Solve y = ±y-σy 3 , ± > , σ > 0. 6. Obtain a continuous solution to the following linear initial value problem with a discon-tinuous coeFcient y + 2 y = g ( x ) , y (0) = 0 , g ( x ) = ( 1 , ≤ x ≤ 1 , , x > 1 ....
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ps1 - u = y/x to solve the following homogeneous equations...

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