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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