March 2, 2009
MathExcel Worksheet # 16: Separable Differential Equations
1. Solve the following differential equations.
√
x
0
(a) y = y
e
(b) (x2 + 1)y 0 = xy
(c)
dy
dx
+ ex+y = 0
2. Solve the following initial value problems.
(a) xy 0 + y = y 2 and y(1) = −1
(b) y 0 tan x = a + y, y(π/3) = a, and 0 < x < π/2.
(c)
dL
dt
= kL2 ln t, and L(1) = −1.
3. Solve the differential equation y 0 = x + y by making the change of variables u = x + y.
4. Solve the differential equation xy 0 = y+xey/x by making the change of variable u = y/x.
5. We’ve done many problems with Newton’s Law of cooling but have not yet solved
the associated differential equation. Formulate Newton’s law of cooling as an initial
value problem (T (0) = t0 ), solve the differential equation, and describe the asymptotic
behavior of the solution (that is, describe what happens to your solution as t → ∞).
6. An object in free fall near the Earth’s surface is acted upon by two forces: gravity and
drag (air resistance). In some cases, it is reasonable to assume that the drag force is
directly proportional to the velocity of the object.
(a) Suppose the initial velocity of the object is 0 m/s. Set up an initial value problem
whose solution is the velocity function of the object during its descent. You may
assume that acceleration due to gravity is 9.8 m/s2 .
(b) Solve the initial value problem from part (a).
(c) If the object is allowed to fall a large distance, it may be better to assume that the
drag force is proportional to the square of velocity Reformulate the initial value
problem in part (b) with this new assumption.
(d) Solve the new initial value problem.
7. A small country has $ 10 billion in old paper currency in circulation. Each day 1 %
of the currency in circulation enters the country’s banks. The government wants to
replace all of the old currency currently in circulation with fresh currency. To do this,
they replace any old currency which enters the banks with the new currency.
(a) Formulate an initial value problem that models this situation.
(b) Solve this initial value problem.
(c) Determine how long it will take for 90 % of the country’s currency to be in the
form of new currency.
8. A vat with 500 gallons of beer contains 4 % alcohol by volume. Beer with 6 % alcohol
is pumped into the vat at a rate of 5 gal/min and the mixture is pumped out at the
same rate. What is the percentage of alcohol after one hour?