MATH 364A: Ordinary Differential Equations (Midterm 1)
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Question 1 (40 points) Solve the following initial value problems.
(a) y + t3 y = t3

y(0) = 0.

(b) y = − (1+x) y y(−1) = 1.

Question 2 (40 points) Solve the second-order initial value problem
2y − 3y − 5y = 0

y(0) = 0

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y (0) = 1.

Question 3 (40 points) For each equation below, first determine whether the equation is exact or not exact. If the equation is exact, find the solution.
(a)
cos(y) + (2y − x sin(y))

dy
= 0. dx (b)
3xy 2 + (y 3 + 3x2 )

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dy
= 0. dx Question 4 (40 points) Consider the following first-order differential equation y = (y 2 − 1)(4 − y 2 ).
Find all critical (equilibrium) solutions and classify their stability.

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Question 5 (40 points) (True/False) In each of the following, determine whether the given function solves the given differential equation or initial value problem.
(a) If k > 0 denotes any real constant, then the function y(t) = e−kt solves the initial value problem y = −ky

y(0) = 1.

(b) The function y(t) = (1 − t)−1 = 1/(1 − t) solves the initial value problem y = y2

y(0) = 1.

(c) The function y(t) = e2t solves the second-order initial value problem y = 2y

y(0) = 1

y (0) = 2.

(d) Any function y(x) defined by the implicit equation x4 + 2x2 y 2 + y 4 = C solves the differential equation x3 + xy 2 + (x2 y + y 3 )y = 0.

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Question 6 (Extra Credit, 10 points) An object with mass m is thrown vertically upward with initial velocity 20m
(i.e. 20 times the mass m) meters per second. Assume the force due to air resistance has magnitude 2|v| directed opposite to the direction of velocity. If the object takes t∗ = 10 seconds to reach its maximum height, what is the mass of the object?

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