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Differential Equations test one
Monday, April 25, 2005
please show your work in order to get full credit
1. Which of the following differential equations is linear?
(a)
d2 x
= − sin x
dt2
(b) x2 y 0 −(tan x)y = ex
(c)
dP
= P − P2
dt
2. For which values of t is the solution to the following differential equation guaranteed to
exist?
(t − 2)(t + 1)y 0 + ty =
3. Given the initial value problem



dy
dx


√
25 − t and y(4) = −2
= 4x −
2
y
y(0) = 1





(a) Is a unique solution to this initial value problem guaranteed to exist near x = 0?
(b) Use Euler’s method with step size h = 0.1 to fill in the following table:
x y
0 1
0.1
dy
dx
approximate ∆y
(c) Write down an equivalent integral equation.
(d) In the method of successive (Picard) approximations to the solution, a first (constant)
solution of y0 (t) = 1 is chosen. What is the next approximate solution?
4. Solve each of the following differential equations or initial value problems.
(a)
dv
4x
=−
dx
v
(b)


dy
2

 x dx − 2y = 4x 



y(1) = 5


(c)


2
3
2

 (12x − 16xy)dx + (4y − 8x )dy = 0 



y(1) = 1
5. Given the equation dy
= −y(y − 2)(y − 99), find each equilibrium (constant) solution and
dt
discuss the stability of each equilibrium solution.
continue to page two


page two
6. A large 5000 liter tank contains 3000 liters of pure water at time t = 0. Beginning at
time t = 0, salt water is pumped into the large tank at a rate of 20 liters per second. Each
liter of this salt water contains 30 grams of salt.
Let S(t) represent the amount of salt in the large tank at time t.
(a) For the first 100 seconds, the large tank is filling up. Write a differential equation for
S(t) when 0 ≤ t < 100, and solve it.
(b) After the first 100 seconds, the tank overflows. Write a differential equation for S(t)
for t ≥ 100, and solve it.