Find a particular solution to

y''+6y'+9y=(-18.5e^(-3t))/(t^2+1)

Test the set of polynomials for linear independence. If it is linearly dependent, express one of the polynomials as a linear combination of the others. (If the set is linearly independent, enter INDEPENDENT. If the set is linearly dependent, enter your answer as an equation using the variables f, g, h, and j as they relate to the question.)

{f(x) = 1 − 7x, g(x) = 9x + x² − x³, h(x) = 1 + x² + 7x³, j(x) = 9 + 7x + 9x³} in ?₃

Let X be a compact space and let Y be a Hausdorff space. Let f ∶ X → Y be continuous. Show that the image of any closed set in X under f must also be closed in Y .

dx dt =ax+by dy dt =−x − y,

2. As the values of a and b are changed so that the point (a,b) moves from one region to another, the type of the linear system changes, that is, a bifurcation occurs. Which of these bifurcations is important for the long-term behavior of solutions? Which of these bifurcations corresponds to a dramatic change in the phase plane or the x(t)and y(t)-graphs?

1. (Euler’s method) First, work out the first three steps by hand. Then approximate y(2) for each of the initial value problems using Euler’s method, first with a step size of h = .1 and then with a step size of h = .05 using the Excel spreadsheet. (a) dy dx = 2xy, y(0) = 1 (b) dy dx = x − y x + 2y , y(0) = 1 (c) dy dx = y + x, y(0) = 1 ignore the excell sheet portion

Let t be a positive integer. Prove that, if there exists a Steiner triple system of index 1 having v varieties, then there exists a Steiner triple system having v^t varieties

Recall that (a,b)⊆R means an open interval on the real number line:

(a,b)={x∈R|a<x<b}.

Let ≤ be the usual “less than or equal to” total order on the set

A=(−2,0)∪(0,2)

Consider the subset

B={−1/n | n∈N, n≥1}⊆A.

Determine an upper bound for B in A. Then formally prove that B has no least upper bound in A by arguing that every element of A fails the criteria in the definition of least upper bound.

Note:

least upper bound is an upper bound for B⊆A that is less than every other upper bound

Classify all the groups of order 175 up to isomorphism.

Solve using laplace transform

y" + 3y = -48t^2e^3t ; y(0) = 2 , y(0) = 1

y" + 6y' + 5y = t - tu(t-2); y(0) = 1 , y'(0) = 0

in the book, Legendre polynomials are obtained to the
degree 5. Normalize polynomials P4 and P5 so that
P4(1) = 1 and p5(1) = 1. This is standard normalization
and these will be standard Legendre polynomials.

Use the recurrence relation for standard Legendre polynomials
to find two more (standard) Legendre polynomials.
Show your work for credit.

Hint: The recurrence relation:

(n+1) P_{n+1}(x) = (2n+1) P_{n}(x) - n P_{n-1}(x)}

Evaluate each integral in and explain why you used the method you did. ∫ ?¹¹ ??? ( ?/z³ ) ??, where ? is the perimeter of a square with vertices?₁ =?+?, ?₂ =−?+?, ?₃ =−?−?, and ?₄=?−?.

Suppose a lady bug is flying about 3-space and now is at the point (0,2,3). Say the temperature at each point is given by T(x,y,z) = x^3 + z^2 - 2ye^x.

a) What direction should the lady bug fly in order to lower her temperature the fastest?

b) What direction could the lady bug fly in order to keep her temperature constant?

Please make a little adjust for the following script then make it work for

Runge Kutta method

Consider the initial value problem

du/dt=t^5   u(0)=0

0<=t<=1

Q1. how we change the below script to make it to solve the above IVP and the answer should be near 4

here is the script

f =@(t ,y)(t^5); % define f by f(t,y)=y^5
for k =1:10
[ tlist , ylist ]= RKfour174 (f ,0 ,1 ,0, 2^k);
error1 (k)= max ( abs ( ylist - ( tlist )));
hlist (k) =1/2^k;
end
x= log ( hlist );
y= log ( error1 );
polyfit (x ,y ,1)

function [tlist,ylist] =RungMethod(f,t0,tfinal ,y0,N)     what is the input for N, n is a stepsize