MIT OpenCourseWarehttp://ocw.mit.edu 18.034 Honors Differential Equations Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.18.034 Recitation: February 12th, 2009 1. Suppose y0(x) is a solution to the equation ′ y + 2xy = f(x) with y0(0) = 3. Find a solution y1(x) with y1(0) = 1. 2. (Birkhoff-Rota, p. 17, #5) For what pairs of positive integers n, r is the function fn(x) = |x|n of class Cr? 3. Show that the equation (3e 2yx 23 − x)y ′ = 1 becomes an equation of B e rnoulli type if x and y are interchanged. Solve that equation and obtain an equation for x. Find an explicit formula for y = y(x) for the solution satisfying y(1) = 0. 4. Find equations for the family of curves ortho gonal to the curves xy = c. 2 Do the same for the fa milies y = cex . 5. Suppose that y is a solution to (x 2 + 1)y ′ + y 2 + 1 = 0 with y(0) = c > 0. Prove that y is decreasing and limxց−1/c y(x) = ∞.
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