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MATH 348 - Sample Test I
September 11, 2015
There is no job so simple that it cannot be done wrong.
1. Answer each of the following T (true) or F (false).
(a)
T
(b)
T
(c)
T
(d)
T
has a
dy
+ y = 0.
dx
dy
2
F
y = e−x is a solution to 2
+ xy = 0.
dx
d2 y
F
The equation
+ xy 2 = 1 is linear.
dx2
dy
F
The solution to the problem
+ x3 y = cos(2x) subject to y(0) = 14
dx
unique solution.
F
y = e−x/2 is a solution to 2
Now assume Ly(x) = y 00 (x) + a1 (x)y 0 (x) + a0 (x)y(x).
(e)
(f)
It is possible for y1 (x) = ex , y2 (x) = e2x and y3 (x) = e3x to be
T F
solutions to Ly(x) = 0.
T
F
If a1 (x) = 0 and a0 (x) = 0 then the solution of Ly(x) = 0 is c1 + c2 x.
2. Let f1 (x) = x and f2 (x) = |x| on the interval I = [−1, 1].
(a) Show that the functions f1 (x) and f2 (x) are linearly independent on the interval I.
(b) Find the Wronskian f1 (x) and f2 (x) on I.
(c) Do the results of (a) and (b) contradict Theorem 7 page 513 of your text?
3. The function yh (x) = c1 e2x + c2 e−2x is the homogeneous solution to the ode
y 00 (x) − 4y(x) = 0.
(a) Find a particular solution to
y 00 (x) − 4y(x) = 6x2 − 8.
Hint: Set yp (x) = Ax2 + Bx + C and use the ode to find A, B and C
(b) Write down the general solution to y 00 (x) − 4y(x) = 6x2 − 8.
5
1
(c) Find the unique solution that satisfies y(0) = and y 0 (0) = .
2
2
4. The equation
sin(x)y 00 (x) − 2 cos(x)y 0 (x) − sin(x)y(x) = 0
models Synchronized Equi-Reciprocal Osmosis or (SERO). Show that y1 (x) = cos(x) is
a solution. Find a second independent solution. Hint:
Z
upon setting c = 0.
tan2 (x) = tan(x) − x + c = tan(x) − x