Description
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.