Find the general solution to the following equation:

y''' + 3y''−4y'−6y = cos(t)

If a population with harvesting rate h is modeled by dx/dt = 9-x^2-h. Find the bifurcation point for the equation.

Let G be a graph. For each of the following, determine if the statement is true or false. If it's true, provide a proof and if it's false, provide a counterexample.

(a) G has a cycle if and only if G has a circuit

(b) G has a closed walk if and only if G has a circuit

(c) G has an odd-lengthed cycle if and only if G has an odd-lengthed circuit

(d) G has an odd-lengthed closed walk if and only if G has an odd- lengthed circuit

Prove Taylor's Theorem (using n=3 case)

Prove f: Zmod p---> Z mod p is a ring homomorphism and find the kernel of f.

Find all possible non-trivial homomorphism

a) S₃ --> Z₆

b) D₄ --> Z₄

c) Z₅ --> Z₁₀

d) Z₁₀ --> Z₅

Find polynomial of degree at most 1 best approximates function f(x) = e^x in the sense of uniform approximation on [0,1]. give the error of approximation of the function f with this polynomial.

Consider the IVP x' = t^2 +x^2, x(0) = 1. Complete the following table for the numerical solutions of given IVP with step-size h = 0.05.

t - x by Euler’s Method - x by Improved Euler’s Method

0 -    1 - 1

0.05 - …….    - ……...

0.1 -    ……. - ……..

(Symmetric Matrices and Quadratic Forms) Examine the following functions for relative extremum values: (a) F = x2y + 2x2 − 2xy + 3y2 − 4x + 7y (b) F = x3 + 4x2 + 3y2 + 5x − 6y

Wе sаw in сlаss thаt in bipаrtitе grаphs thе mаximum mаtсhing аnd minimum vеrtеx сovеr hаvе thе sаmе sizе. (Thе numbеr of еdgеs in thе mаtсhing еquаls thе numbеr of vеrtiсеs in thе сovеr.)

(а)Find аn еxаmplе of а non-bipаrtitе grаph in whiсh thе minimum vеrtеx сovеr is еxасtly twiсе аs lаrgе аs thе mаximum mаtсhing.

(b)Find аnothеr еxаmplе of а non-bipаrtitе grаph in whiсh thе minimum vеrtеx сovеr аnd thе mаximum mаtсhing hаvе еquаl sizеs.

Find y as a function of x if

y(4)−12y‴+36y″=−512e^-2x

y(0)=−1,  y′(0)=10,  y″(0)=28,  y‴(0)=16

y(x)=

A mass of 2 kg is suspended from a spring with known spring constant of 10N/m and allowed to come to rest. it is then set in motion by giving it an initial velocity of 150 cm/sec. Find an expression for the motion of the mass, assuming no air resistance.

If X is any topological space, a subset A ⊆ X is compact (in the subspace topology) if and only if every cover of A by open subsets of X has a finite subcover.