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1) Determine the values of m, for which y = e

mx is a solution of the

differential equation

y

00 − 5y

0 + 6y = 0.

If y1 and y2 are solutions to the differential equation above and c1 and

c2 are constants, is y = c1y1 + c2y2 also a solution? Why or why not?

2) Find the 1 parameter family of solutions to

y

0 = (x + 5)(x − 3)−1

(x + 1)−1

.

Show which values of xo ∈ R guarantee existence and uniqueness of

the solution to the IVP y(xo) = yo by invoking an appropriate theorem

from the text.

3) When an object at room temperature is placed in an oven whose temperature is constant at Tf , the temperature of the object will increase

with time, approaching the temperature of the oven. The temperature

T of the object is related to time by through the differential equation

T

0 = k(T − Tf )

where k is a real constant.

Given that T(0) = Ti

, use separation of variables to solve this IVP for

T in terms of the independent variable, t, and the constants, k, Tf and

Ti

.

4) Solve by separating variables, the initial value problem

y

0 = xy2

e

x

, y(0) = 2

and comment on uniqueness of the solution.

5) Find a 1 parameter family of solutions to the following first order linear

differential equation,

x

3

dy

dx + x

2

y = x.

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