1) Solve each of the following differential equations.

a)16y"-8y'+y=0

b) (d^4y)/(dx^4)-13((d^2y)/(dx^2))+36y=0

2) use Variation of Parameters to solve y"+16y=(1/3)csc4t

3) use undetermined coefficients to solve y"-5y'+4y=3e^(3t)-5e^(2t) with y'(0)=-1 and y(0)=1

4) Explain why the product (A+B)(A-B) not equal A^2-B^2 fro two NXN matrices A and B. what is the product of (A+B)(A-B)?

Suppose h, k, r, s, t∈Z.Set a=3rt(2s+t) and b=3rs(s+2t).Prove the cubic polynomial

f (x) = (x − h)(x − a − h)(x − b − h) + k
passes through the point (h,k), has integer roots, has local extrema with integer coordinates, and has an inflection

point with integer coordinates.

f(x) = ((x − 1)^2) e^x

How easy would it be to apply the Bisection Method compared to Newton's method and modified Newton's method to the function f(x)? Explain.

Let a and b be integers which are not both zero.

(a) If c is an integer such that there exist integers x and y with ax+by = c, prove that gcd(a, b) | c.

(b) If there exist integers x and y such that ax + by = 1, explain why gcd(a, b) = 1.

(c) Let d = gcd(a,b), and write a = da′ and b = db′ for some a′,b′ ∈ Z. Prove that gcd(a′,b′) = 1.

In Matlab (diary on), do the following: 

1. Generate N+1=11 equi-spaced nodes Xi in the interval [−5,5]:
         X = [-5:1:5];    (to see values, omit the ;)
   or    X = linspace(-5,5,11);
   and evaluate f(x) at these nodes:
         Y = 1./(1+X.^2);
   The N+1 points (Xi, Yi) are the data points to be interpolated by various methods. 
   Plot them to see where they are:
         plot(X,Y,'o')
         title('N+1 = 11 equi-spaced points of Runge function')
         pause 

   Also generate lots of points xi at which to evaluate f(x) and the interpolants for plotting:
         x = [-5:0.1:5];   (this is a lower case x, not X)
   Evaluate f(x) at these xi's and plot y=f(x) and the data points:
         plot(x,y,'-', X,Y,'o')
         title('Runge f(x) and data pts')
         pause 

Now, we use the data points (Xi, Yi) to construct various interpolants.
A good interpolant should "reproduce" the function f(x) as close as possible. 
Let's try a few.


2. Nth degree interpolating polynomial:
   Use Matlab's polyinterp to construct (the coefficients of) the Nth degree interpolating polynomial (here N=10):
         pN = polyfit( X,Y, N); 
   Now this can be evaluated anywhere in the interval [-5,5] with polyval, e.g. at the xi's:
         v = polyval( pN, x); 
   Find the Inf-norm error ∥y-v∥∞:
         err = norm(y-v, inf) 
   and plot both f(x) and pN(x) on the same plot:
         plot(x,y,'-b', x,v,'*-r')
         title('f(x) and pN(x) at plotting pts')
         pause 

   Is this a good interpolant ?  Why ?
        

   
3. Interpolation at Chebychev nodes:
   Generate N+1=11 Chebychev points (Xchebi, Ychebi) in [a,b]:
         fprintf('------ chebychev nodes ------\n')
         K = N+1;
         a=−5;  b=5;
         for i=1:K
             Xcheb(i)=(a+b)/2 + (b−a)/2 *cos( (i−0.5)*pi/K );
         end
         Ycheb = 1./(1+Xcheb.^2);

   Follow the steps in 2. to produce the Nth degree interpolating polynomial pNcheb 
   based on the Chebychev nodes, its values vcheb at the xi's, the error ∥y − vcheb∥∞,
   and plot both f(x) and pNcheb(x) on the same plot.

   Compare the error and plot with those from 2.
   Which one is better ?  why ?
        


4. Piecewise linear interpolation:
   Use Matlab's interp1 to construct the linear interpolant:
         lin = interp1(X,Y, x, 'linear'); 
   Repeat the steps of 2. 

   Compare errors and plots.
        

5. Piecewise cubic interpolation:
   Use Matlab's interp1 to construct the cubic interpolant:
         cub = interp1(X,Y, x, 'cubic'); 
   Repeat the steps of 2. 

   Compare errors and plots.
        


6. Cubic spline interpolation:
   Use Matlab's interp1 to construct the spline interpolant:
         spl = interp1(X,Y, x, 'spline'); 
   Repeat the steps of 2. 

   Compare errors and plots.
        


7. To see that the error gets worse for bigger N for equi-spaced nodes but not for Chebychev nodes 
   (for this f(x) at least), repeat 2. and 3. with N = 20.

Let U = {A ∈ Mat(2; ℚ) : AB = BA for all B ∈ Mat(2; ℚ)}.

(i) Show that  U is a subspace of Mat(2; ℚ).

(ii) Show that E ∈ Mat(2; ℚ) is a basis of U. (E: identity matrix)

(iii) Find the complement for U

5. Show that if R is a division ring,then Mn(R) has no nontrivial two-sided ideals.

Need the detailed calculation process about why at the same BER condition, BPSK is 3dB more power efficient that BFSK

Suppose U is a subspace of a finite dimensional vector space V. Prove there exists an operator T on V such that the null space of T is U.

Please answer question (b)

9. (a) Show that the PDE Ux = 0 has no solution which is C1 everywhere and satisfies the side condition u(x,x^3) = x , even though the side condition curve y = x^3 intersects each

characteristic line (y = d) only once.
(b) Part (a) demonstrates the necessity of the transversality condition on the intersections of the side condition curve with the characteristic lines. Explain why.

Hint. At what angle does the curve y = x^3 meet the x-axis?

Find the unique solution u of the parabolic boundary value problem

Ut −Uxx =e^(−t)*sin(3x), 0<x<π, t>0,

U(0,t) = U(π,t) = 0, t > 0,

U(x, 0) = e^(π), 0 ≤ x ≤ π.

can you make me a sample problem of 2 geometric sequence, harmonic sequence, arithmetic sequence so that's all 5 this is for Grade 10 Mathematics all with answers and solutions thankyou