Texas A&M University Department of Mathematics Volodymyr Nekrashevych Fall 2011 MATH 308 Homework 1 1.1. Sketch a direction field for the differential equation y 0 = 1 + 2y. Based on the direction field, determine the behavior of y as t → ∞. If this behavior depends on the initial value of y at t = 0, describe this dependency. 1.2. Do the same as in the previous problem for the equation y0 = y2. 1.3. A spherical raindrop evaporates at a rate proportional to its surface area. Write a differential equation for the volume of the raindrop as a function of time. 1.4. Consider the differential equation dy/dt = −ay + b, where both a and b are positive numbers. Solve the differential equation. Sketch the solution for several different initial conditions. Describe how the solutions change under each of the following conditions 1. a increases. 2. b increases. 3. Both a and b increase, but the ration b/a remains the same. 1.5. Find the general solution of the differential equation y 0 − 2y = t2 e2t and determine how solutions behave as t → +∞. 1.6. Consider the initial value problem 1 y 0 + y = 2 cos t, 2 y(0) = −1. Solve it, and then find the coordinates of the first local maximum point of the solution for t > 0.