1)Find the power series solution for the equation y'' − y = x

2)Find the power series solution for the equation y'' + (sinx)y = x; y(0) = 0; y'(0) = 1

Provide the recurrence relation for the coefficients and derive at least 3 non-zero terms of the solution.

Find a particular solution to the following non homogenous equations

1) y''' + y = t^3 + sin (t) + 11e^t

2) y'' + y = 2tsin(t)

3) y''''' - 4 y''' = e^2t + t^2 +5t + 4

Solve the initial value problem

11. xdx−y2dy=0, y(0)=1

12. dydx=yx, y(1)=−2

16. dydx=sinxy, y(0)=2

17. xy′=√1−y2, y(1)=0

23. Mr. Ratchett, an elderly American, was found murdered in his train compartment on the Orient Express at 7 AM. When his body was discovered, the famous detective Hercule Poirot noted that Ratchett had a body temperature of 28 degrees. The body had cooled to a temperature of 27 degrees one hour later. If the normal temperature of a human being is 37 degrees and the air temperature in the train is 22 degrees, estimate the time of Ratchett's death using Newton's Law of Cooling.

(abstract algebra)

(a) Find d = (26460, 12600) and find integers m and n so that d is expressed in the form m26460 + n12600.

(b) Find d = (12091, 8439) and find integers m and n so that d is expressed in the form m12091 + n8439.

1. Explain the outcome of 3^4^5. In particular, what is the order of execution of the two exponentiation operations?

2. Write (5^4^3)−1 as a product of prime numbers.

3. The greatest common divisor of two integers a and b can be written as a linear combination (with integer coefficients k and ℓ) of a and b: gcd(a,b)=ka+ℓb.

   In Sage this is achieved with the command xgcd. Look in the help page of this command to write the greatest common divisor of 12214 and 2012 as an integer linear combination of these two numbers.

   Use Sage to verify your result.

4. What is the difference in Sage between 1/3+1/3+1/31/3+1/3+1/3 and 1.0/3+1.0/3+1.0/31.0/3+1.0/3+1.0/3? Explain.

1. Give your own example of a plane figure and its quadrature.
2. How does the quadrature of the triangle depend on the quadrature of the rectangle?
3. Draw a lune like Hippocrates.’ What is the relationship between the large semicircle and the semicircle that contains the lune?

"We want to verify that IP(·) and IP^-1(·) are truely inverse operations. We consider a vector x = (x1, x2, . . . ,x64) of 64 bit. Show that IPfive bits of x, i.e. for xi, i = 1,2,3,4,5.

Sketch the region of continuity for f (x; y) on a set of axes and sketch the region of

continuity for df/dy (x. y) on a separate set of axes. Apply Picard’s Theorem to determine whether the

solution exists and whether it is unique.

a) y'  = 2x²y + 3xy² ; y(1) = 2

b) y' = sqrt(2x - 3y) ; y(3) = 2

Let S be a subset of a vector space V . Show that span(S) = span(span(S)). Show that span(S) is the unique smallest linear subspace of V containing S as a subset, and that it is the intersection of all subspaces of V that contain S as a subset.

1.let {v=(1,2,3,5,9),v2=(3,1,2,8,9),v3=(2,-5,5,9,4)} and {u1=(0,1,1,1,2),u2=(0,2,-2,-2,0)} be basis of subspaces V and U of R5 respectively.find a basis and the dimension of V+U and V intersection U.

2.does a matrix have a right inverse ?if so find one A=[2,-3,-7,11;3,-1,-7,13;1,2,0,2]

3.find the interpolating polynomial that passes through the point (1,2),)(-1,-8) and (2,1)

Consider the equation uux + uy = 0 with the initial condition

u(x, 0) = h(x) = ⇢ 0 for x > 0

u_o for x < 0,   with u_o< 0.

Show that there is a second weak solution with a shock along the line x = u_o y / 2

The solution in both mathematical and graphical presentation before and after the shock.

Find the adjoint of matrix A, the determinant of matrix A, and the determinant of the adjoint A.

A= 1 1 0 2

2 1 1 0

0 2 1 1

1 0 2 1

A SEIRS model with stochastic transmission :project proposal

National governments issue debt securities known as sovereign bonds, which can be denominated in either local currency or global reserve currencies, like the U.S. dollar or euro. First define what these bonds are. Why are these issued? Then discuss the issues that can arise when investors invest in these types of bonds. What are the advantages and disadvantages of these bonds? Are there unique issues that can arise only with this type of bond? Would you invest in sovereign bonds?

Show that radical 3, radical 5, radical 7, radical 24, and radical 31 are not rational numbers