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 5th, 2009 1. Consider the equation y ′′ + y ′ = 2y. For what a is eay a solution? Find a ′′′′ ′′′ ′′ solution y0 to y + y = 2y with exponential g rowth as x → −∞ and limx→∞ y0(x)/x = 1. 2. Suppose that if y0 is a solution to y ′′ + y ′ − 2y = F (x) on all of (−∞, ∞), then there is another solution y1 to the same equation with |y1(x) − y0(x)| → 0 as x → ∞. x3. Show that y = e and y = cos x cannot be solutions of the same first-or der ′equation y = f(x, y) on any interval containing the origin. 4. Solve ydx + 3xdy = 14y4dy. 5. Suppose that a trajectory of (3x2 − y)dx + (3y2 − x)dy = 0 contains the point (1, 1). Show that it also contains the points (1, −1), (−1, 1), (0, 1), (1, 0). ′6. (Birkhoff-Rota: p. 6, # 7) Show that solutions of y = g(y) are convex up or convex down for given y according as |g| is an increasing or decreasing function there.
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