MATH 308 Homework 1

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
y 0 + y = 2 cos t,
y(0) = −1.
Solve it, and then find the coordinates of the first local maximum point of the solution
for t > 0.