Question

Find all solutions to the following equations: (a) z^6 = 1 (b) z^4 = −16 (c) z^6 = −9 (d) z^6 − z^3 − 2 = 0

Answer #1

Find all z ∈ C which satisfy the following polynomial equations:
(a) z^3 + 27 = 0. (b) 2z^16 − 1 = 0

Find the general solutions for the following differential
equations:
a. (D^2 - 8D + 16)y = 0
b. (D^4 - 9D^3)y = 0

Find all solutions to the given equations. Then find all
solutions in the interval [0,2pie)
a. 2sin^2(theta)-sin(theta)-1=0
b. tan(2theta)=cot(2theta)
c. sin(3theta)-sin(6theta)=0
d.sec^2(theta)=2tan(theta)
Please Show All Work

1.
a) Draw a sketch of: {z∈C|Im((3−2i)z)>6}.
b) If 2i is a zero of p(z)=az^2+z^3+bz+16, find the real
numbers a,b.
c) Let p(z)=z^4−z^3−2z^2+a+6z, where a is real. Given that 1+i
is a zero of p(z):
Find value of a, and a real quadratic factor of p(z). Express
p(z)as a product of two real quadratic factors to find all four
zeros of p(z).

For each part, find all possible solutions to the given
equations (if any exist).
i ) x = 7 mod 9, x =3 mod 4
ii) 3x^2 +1 = 0 mod 10
iii) 3x+7 = 9 mod 10

(Part 1) Find all of the solutions of the given differential
equations:
a.) y' = -2y (answer should be y = -(1 / ln2) * ln(t
* ln(2) + c))
(Part 2) Find the solution of the IVP:
b.) y' = -2y3, y(0) = 0
c.) y' = 1 + cos(y), y(0) = pi / 2 (answer should be y(t) =
2arctan(1 + t))
d.) y' = sqrt(1 - y2), y(0) = 0 (Hint: y' > 0)
Please show work!

Find the explicit descriptions (“solutions”) for the following
difference equations: ??+1 = 2 ?? −1, ?0 = 3 ??+1 = 3 ?? + 2, ?0 =
3 ??+1 = 0.3 ?? − 3.2, ?0 = 0

for 10-12 you will solve the following system of equations:
2x+y+z=-2 2x-y+3z=6 3x-5y+4z=7 10) what is the solution for x? a)2
b)-3 c)infinitely many solutions d)no solution 11) what is the
solution for y? a)2 b)0 c)inifinitely many solution d)no solution
12) what is the solution for z? a)4 b)-8 c)infinitely many
solutions d)no solutions

Find the general form of all solutions (if there are any) for
the following equations.
(a) 172x + 20y = 1000
(b) 18x + 5y = 48
(c) 158x − 57y = 7
(d) 6x + 51y = 22

Find all solutions to the differential equations.
(a) x^2 yy' = (y^2 − 1)^(3/2)
(b) y' = 6xe^(x−y)
(c) y' = (2x − 1)(y + 1)
(d) (y^2 − 1) dy/dx = 4xy^2
Leave your answer as an implicit solution

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