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1) a) State sufficient conditions for the existence and uniqueness of a
solution on some interval, I, including the point xo of the IVP
dy
dx = f(x, y) with y(xo) = yo
where xo, yo ∈ R.
b) For what values of xo, yo ∈ R is the IVP
dy
dx = y
2/3
√
x − 2 with y(xo) = yo
guaranteed to have a unique solution on some interval containing
the point xo?
2) Find an implicit family of solutions to
dy
dx =
x
3
cosy
3) Find the general solution to the IVP (i.e. find the 1 parameter family
of solutions to the IVP)
dy
dx + 2xy = x
3
4) Verify that the following differential equation is exact (you will want
to quote a theorem from the text, which you should know ,in order to
verify this) and then find a family of implicit solutions.
(3x
2
y + 8xy2
)dx + (x
3 + 6y
2 + 8x
2
y)dy
5) Solve the homogeneous differential equation
(y
2 + xy)dx − x
2
dy = 0
using the substitution y = ux.
6)
y
2
dx + (4xy + 1)dy = 0
is a nonseparable differential equation which can be made exact by
multiplying through by an appropriate integrating factor. Find the
integrating factor, verify that the new equation is in fact exact and
then find a family of implicit solutions.
7) Identify the type of D.E. (Linear, Bernoulli, Exact, Homogeneous, Separable)
a)
dy
dx + 3x
4
y = xy3
b)
(x
3 + xy2
)dx + (y
3
)dy = 0
c)
dy
dx + (x
3
sin(x))y = e4x
d)
(e
2y − y)cos(x)dy
dx = ey
sin(x)
e)
12x
3
y
5
dx + (15x
4
y
4 + 6y)dy = 0
8) Consider the autonomous differential equation:
dy
dx = (y
3 + y
2 − 12y)e
y
Find and classify the equilibrium solutions and draw the 1 dimensional
phase portrait.
9) Suppose the rate at which a student complete his or her homework is
proportional to the amount of homework he or she has remaining. If
h(t) represents the amount of homework remaining, write a differential
equation which describes the system. If the student begins with Ho
amount of homework, and after 50 hours has 4Ho
5
remaining, find an
explicit solution for h(t) for t > 0. How long will it take for the student
to finish 9
10 of the total homework?
2
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