Math 308
Due: Feb 15th
Spring 2016
1. Consider the Autonomous Differential Equation
y 0 = cos y − y 2
(a) Plot the function f (x) = cos x − x2 on the interval [−2, 2].
(b) Find the stationary solutions to the autonomous equation.
(c) For what range of y-values is the solution increasing/decreasing?
(d) Show (by hand) that
y 00 = −y 0 (sin y + 2y)
(e) For what range of y-values is the solution concave up/down?
2. The Newton’s have taken out a loan to buy a house. The loan is for 200,000
dollars with an annual interest rate of 5%, a 50,000 dollar down payment and
a 25 dollar a month service fee. If the Newton’s pay 850 dollars a month when
will they be finished paying off the loan?
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