Posted: November 20th, 2015

Mathematics

Mathematics

1. (Do not find the solution.) Sketch a direction filed of
dy
dx = x – 2y
Show, at least, the slope arrows at (0, 1), (1, 1) and (1, 0).
Draw the likely solution curve at
y(0) = 1.
2. Find the critical points and phase portrait. Classify each
critical point
y
‘ = (y – 1)(3 – y).
3. Solve the following two first-order equations:
(1) dy
dx =
3x
2 – 2x
cos(y + 1),
(2) xy’ – 4y = 4x
2
.
4. Find yc (the general solution of the homogeneous equation)
and write down only the yP form in the method of undetermined
coefficients.
y
” – 4y
‘ + 13y = e
3x + cos 3x + x + e
2x
sin 3x.
5. Find the general solution. Here you need to find yp twice,
by both (1) the method of undetermined coefficients and
(2) the method of variation of parameters.
4y
” – y
‘ = 4
6. (1) Show S = {f(x) | f(1) = 2} is not a subspace of C[0, 1].
(2) Show S = {

a b c
T
| a · b · c = 0} is not a subspace
of R3
.
7. Solve the linear system by the method of Gauss-Jordan
elimination (must reach Reduced Row Echelon Form)
+x2 +x4 = 1
x1 +x2 -x3 +x4 = 1

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