y'' − 3y' + 2y =− x sin x
STEP BY STEP CAREFULL INSTRUCTION.
In: Advanced Math
Find particular solution to the ODE: y"(x)+y(x)=4xcosx
In: Advanced Math
1)
At the end of the following algorithm segment, what value is in variable n?
n:= 31 WHILE ( n is prime) n:= n -2 END WHILE
2) Prove that there is no greatest negative real number using contradiction or contraposition.
In: Advanced Math
Let A := 1+√5 2 and B := 1−√5 2 . These are both roots of the equation x2 = x + 1. Show that fn = An −Bn √5 , where fn is the nth fibonacci number.
In: Advanced Math
The students in a Math class decide to divide themselves into groups of equal size to study multiple congruencesWhen they try groups of 4, 3 are left over; groups of 5, 2 are left; finally they try groups of 7 and discover that if the professor joins in then they can use 7. The professor, reasoning that he never really understood modular arithmetic anyway, thinks it will probably do him some good and agrees to join in. What is the number of students in the class if you are told it is between 150 and 250
In: Advanced Math
Let a, b, c ∈ Q, (a, b) /= (0, 0), and consider the line L = {(x, y) ∈ R2 ; ax + by = c}. State and prove a criterion in terms of the data a, b, c ∈ Q to check whether L∩Z2 = ∅, i.e., the line passes through no point with integer coordinates in the plane. Give an explicit example of such a line that is neither parallel to the x-axis nor to the y-axis!
In: Advanced Math
A polygon is called convex if every line segment from one vertex to another lies entirely within the polygon. To triangulate a polygon, we take some of these line segments, which don’t cross one another, and use them to divide the polygon into triangles. Prove, by strong induction for all naturals n with n ≥ 3, that every convex polygon with n sides has a triangulation, and that every triangulation contains exactly n − 2 triangles. (Hint: When you divide an n-gon with a single line segment, you create an i-gon and a j-gon for some naturals i and j. What does your strong inductive hypothesis tell you about triangulations of these polygons?)
In: Advanced Math
Let S={(x,y,z) | x^2+y^2+4z^2=9} be a closed surface in R3. F(x,y,z)=(cos x, sin x, x^2+y^2+z^2) is a vector field. Compute ∫∫ (▽xF) ds
In: Advanced Math
1. For the following two matrices
A = 1 2
3 4
B = 1 3 5
2 4 6
Find:
a). AB, is AB = BA ?
b). All row and column vectors of A and B
2. Let X1=[1, 0, 0], X2=[0, 3, 0], and X3=[0, 0, 2], use
definition of linear dependency to
show that the above three vectors are dependent or not.
3. Show that the following matrix is singular or nonsingular,
and find A-1 if possible.
A =
1 2 3
3 2 1
2 3 1
In: Advanced Math
Consider the matrix transformation T in R2that does the following actions. It rotates a vector by radians counterclockwise and then reflects the vector across the x axis.
Apply T to i.
Apply T to j.
Apply T to another vector of your choice, but one that has magnitude 1 so it sits on the unit circle.
What is the generalized form of T?
In: Advanced Math
A college student owes $2000 to a credit card company, which charges interest at an annual rate of 10%. The student makes payments continuously at a constant rate of $25/month ($300/year).
a. set up the initial value problem describing the situation.
b. solve the initial value problem from part (a)
c. find the time T it will take to pay off the debit
In: Advanced Math
The following cubic polynomial has three real roots p(x) = 32x3 − 110x2 + 123x − 45
(a) Plot p(x) for .9 ≤ x ≤ 1.6 and indicate the locations of the three roots. What are the exact roots of p(x)?
(b) Write a MATLAB code to run Newton’s method with x0 = 1.2 and discuss the convergence.
(c) Write a MATLAB code to run the secant method starting with x0 = 1.31 and x1 = 1.2 and discuss the convergence.
(d) Write a MATLAB code to run the bisection method with the initial bracket [.9, 1.6] and discuss the convergence.
In: Advanced Math
Find the general solution of the equation
e^(3x)y'' + e^(3x)y' + e^(x)y = 1,
given that y1 = cos(e^(-x) ) is a solution of the corresponding homogeneous equation.
In: Advanced Math
We know that 1 + 2 + 3 = 1 · 2 · 3, that is, there exist three positive integers whose sum equals their product.
Prove or disprove: There exist twenty positive integers (not necessarily distinct) whose sum equals their product.
In: Advanced Math
Find (a) the rank of the matrix below, (b) a basis for its row space, and (c) a basis for the solution space of Ax=0. [To save you time, I include the RREF of A.]
A = {(1,2,-3), (2, -1, 4), (4, 3, -2)}; each triple is a row of A, from left to right (top to bottom).
The RREF of A is {(1,0,1), (0,1,-2), (0,0,0)} (left to right, top to bottom, as above).
In: Advanced Math