Math 331: Differential Equations
Practice Midterm 1
Department of Mathematics and Statistics
University of Massachusetts
Instructor: Alain Bourget February 14, 2005
Question 1: Solve the following initial value problems. Indicate if the equation is linear, separable, exact
or homogeneous.
(a)(ty0+ (1 + t)y=t, t > 0
y(ln 2) = 1 (b)(y0=2 cos 2x
3+2y
y(0) = −1
(c)((x2+ 3xy +y2)dx −x2dy = 0
y(0) = y0
(d)((sin y−x/y)y0= 1
y(1) = π
Question 2: A ball with mass of 0.1 kg is thrown upward with initial velocity 10 m/sec from the roof of a
building of 10 m high. The force due to air resistance is equal to |v|/10 where the velocity vis measured in
m/sec. Find the time that the ball hits the ground. You can give your answer in form of an implicit equation.
Question 3: Consider the autonomous equation given by
dy
dt =k(1 −y)2where k > 0.
(a) Draw the direction fields of y.
(b) What are the equilibrium solutions? Are they stable, semi-stable or unstable?
(c) Without solving the equation, sketch some integral curves of the solutions y(t).
(d) Find the solution of the equation if the initial condition y(0) = y0is given.
