Question

Prove that following equations have no rational solutions.

a) x^3 + x^2 = 1

b) x^3 + x + 1 = 0

d) x^5 + x^4 + x^3 + x^2 + 1 = 0

Answer #1

Solve the following equations algebraically. Be sure to check
for extraneous solutions.
A). (9^2x)(1/3)^(x+2) = 27(3^x)^-2
B). 3^x+4 = 2^1-3x
C). log base 3(x-2) + log base 3 (x-4) =2

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

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

Prove the following theorem about rational numbers:
If [(x, y)] ≠ [(0, 1)] then [(x, y)] has a multiplicative
inverse

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

Prove that all the solutions to the equation x2 = 4x
-3 are rational. Justify your answers.
(subject is computer theory)

Prove that the complete set of solutions to the congruence x
4 − x 2 + x + 3 ≡ 0 (mod 51) is given by {x ∈
Z: x ≡ 36 (mod 51)}

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

2. Solve for the general and definite solutions for the
following difference equations:
yt+1=-0.5yt +3, y0
=1
yt+2= yt+1-2yt+12,
y0=4, y1=5

Which of the following systems of equations have nonzero
solutions? If the solution is not unique, give the set of all
possible solutions.
a)3x+4y=0 b)4x-y=0 c)2x-6y=0
6x+2y=0 4x-y=0 -x+3y=-2

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