Homogeneous Linear Dierential Equations with Constant Coecients
1. (a)
y 3y + 2y = 0 = m2 3m + 2 = (m 1) (m 2) = 0 = m = 1 or m = 2
Hence,
y = c1 ex + c2 e2x .
(b)
y y = 0 = m2 m = m (m 1) = 0 = m = 0 or m = 1
Hence,
y = c1 + c2 ex .
(c)
y + 2y + 10y = 0

Systems of Homogeneous Equations
1. (a) The reduced system shows that it has unique solution, i.e., trivial solution.
(b) The two equations are scalar multiple of each other, so the system has only one
equation for two unknowns, i.e., non-trivial solution

Separable Equations
1. (a)
dy
3x2 + ex
=
= 4y 3 dy = 3x2 + ex dx =
dx
4y 3
= y 4 = x3 + ex + C
4y 3 dy =
3x2 + ex dx
(b)
x2 + 1
dy
x dx
1
dy
= xy =
=
= ln |y| = ln x2 + 1 + C
dx
y
x2 + 1
2
2 + 1, where A = eC , as A x2 + 1 > 0
= |y| = y = A x
(c)
x2
dy
+

Matrices Operations
1. (a) Since the number of columns in B is not equal to the number of rows in A, so BA
is undened.
(b)
A45 C52 + D42 = AC42 + D42 = (AC + D)42
(c) As A is a 4 5 matrix and E is a 5 4 matrix, so AE is a 4 4 matrix. However,
B is a 4 5 m

Partial Derivatives
Limits
For a single variable function f (x), the limit lim f (x) exists only if the right-hand side limit
xa
equals to the left-hand side limit, i.e.,
lim f (x) = lim+ f (x) .
xa
xa
For a two variables function f (x, y), the limit
lim