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Texas A&M University Department of Mathematics Volodymyr Nekrashevych Fall 2010 MATH 308 Homework 3 3.1. Sketch the graph of f (y) versus y, determine the critical (equilibrium) points, and classify each one as asymptotically stable or unstable. dy/dt = ay + by 2 , a > 0, b > 0, −∞ < y0 < ∞. 3.2. Sketch the graph of f (y) versus y, determine the critical (equilibrium) points, and classify each one as asymptotically stable, unstable, or semistable. dy/dt = y 2 (y 2 − 1). 3.3. Determine whether the equation is exact. If it is exact, then solve the equation. dy/dx = − 4x + 2y . 2x + 3y 3.4. Solve the initial value problem (2x − y)dx + (2y − x)dy = 0, y(1) = 3. 3.5. Find the value of b for which the given equation is exact, and then solve it using that value of b. (ye2xy + x)dx + bxe2xy dy = 0. 3.6. Show that the given equation is not exact but becomes exact when multiplied by the given integrating factor. Solve the equation. x2 y 3 + x(1 + y 2 )y 0 = 0, µ(x, y) = 1/xy 3 .