Consider the initial-value problem y' = 2x − 3y + 1, y(1) = 7. The analytic solution is y(x) = 1/9 + 2/3 x + (56/9) e^(−3(x − 1)).

(a) Find a formula involving c and h for the local truncation error in the nth step if Euler's method is used.

(b) Find a bound for the local truncation error in each step if h = 0.1 is used to approximate y(1.5). (Proceed as in this example.)

(c) Approximate y(1.5) using h = 0.1 and h = 0.05 with Euler's method. (Round your answers to four decimal places.)

h = 0.1          y(1.5) ≈______

h = 0.05         y(1.5) ≈______

(d) Calculate the errors in part (c) and verify that the global truncation error of Euler's method is O(h). (Round your answers to four decimal places.) Since y(1.5) =______, the error for h = 0.1 is E_0.1 = ______, while the error for h = 0.05 is E_0.05 = ______. With global truncation error O(h) we expect E_0.1/E_0.05 ≈ 2. We actually have E_0.1/E_0.05 = _______ (rounded to two decimal places).

1. Find the first four terms in each portion of the series solution around x₀ = 0 for the following differential equation. x² y^// - 5x y^/ + 6y = 0

Differential Geometry

Open & Closed Sets, Continuity

Prove f(t)=(x(t),y(t)) is continuous iff x(t) and y(t) are continuous

(1) Show that the set { 1 m + 1 n : m, n ∈ N} is countable.

(2) Show that the set {a + b √ 2 : a, b ∈ Q} is countable.

(3) Show that the intersection of two countable sets is countable.

(4) Show that the set of all irrational numbers is uncountable.

(5) Let C = {0, 1, 2, . . . , 9}. Show that the set C ×C × · · · is uncountable. [Hint: Imitate the proof we had for E × E × · · ·, where E = {0, 1}.]

Partial Differential Equations

(a) Find the general solution to the given partial differential equation and (b) use it to find the solution satisfying the given initial data.

Exercise 1. 2∂u ∂x − ∂u ∂y = (x + y)u

u(x, x) = e −x 2

Exercise 2. ∂u ∂x = −(2x + y) ∂u ∂y

u(0, y) = 1 + y 2

Exercise 3. y ∂u ∂x + x ∂u ∂y = 0

u(x, 0) = x 4

Exercise 4. ∂u ∂x + 2y ∂u ∂y = e −x − u

u(0, y) = arctan y

Exercise 5. ∂u ∂x+v ∂u ∂y = −ru

(here r and v 6= 0 are constants) u(x, 0) = sin x x

Find the eigenvalues and eigenfunctions for the following boundary value problem.

y"+6y'-(L-8)=0, y(0)=0, y(2)=0 L == Lambda

By sepration of variables Solve

a- 9U_yy=6U_xU_y

b- 4U_xx=6U_xy

Consider the ODE y"+ 4 y'+ 4 y = 5 e^(− 2 x ). (

a) Verify that y 1 ( x) = e − 2 x and y 2 ( x) = xe − 2 x satisfy the corresponding homogeneous equation.

(b) Use the Superposition Principle, with appropriate coefficients, to state the general solution y h ( x ) of the corresponding homogeneous equation.

(c) Verify that y p ( x) = 52 x 2 e − 2 x is a particular solution to the given nonhomogeneous ODE.

(d) Use the Nonhomogeneous Principle to write the general solution y ( x ) to the nonhomogeneous ODE.

(e) Solve the IVP consisting of the nonhomogeneous ODE and the initial conditions y(0) = 1 , y 0 (0) = − 1 .

Describe a problem that you are currently faced with at work or in your personal life that could be solved by using an optimization model. Describe what the problem is, why optimization modeling could help you and how you would approach solving the problem with an optimization model.

a) If you add the terms ax^2+bxy+cy^2 to the function L(x,y) = 1 - 0.5y, how would this affect the derivatives of L(x,y) at (0,0)?

b) What a, b and c you would pick to make those derivatives match the derivatives of the f(x,y) = sqrt(x^2+1-xy-y) at (0,0).

c) Define this new L(x,y).

For this activity, select a recurring quantity from your OWN life for which you have monthly records at least 2 years (including 24 observation in dataset at least). This might be the cost of a utility bill, the number of cell phone minutes used, or even your income. If you do not have access to such records, use the internet to find similar data, such as average monthly housing prices, rent prices in your area for at least 2 years (You must note the data source with an accessible link). Data can also be monthly sales of some particular commodity. 1.4 Please do the descriptive analysis, using the method of index number and Exponential Smoothing individually. And try to explain the pattern you find. 1.5 Use two methods you learned to predict the value of your quantity for the next year (12 months). And make comparison with two results.

Find the best weights (w0...w4) of the highest possible order finite difference formula of the form f'(x) ~ w0*f(x) + w1*f(x+h) + w2*f(x+2h) + w3*f(x+3h) + w4*f(x+4h) and use Taylor series to predict the convergence order as h is decreased.

Prove that 1^3 + 2^3 + · · · + n^3 = (1 + 2 + · · · + n)^2 for every n ∈ N. That is, the sum of the first n perfect cubes is the square of the sum of the first n natural numbers. (As a student, I found it very surprising that the sum of the first n perfect cubes was always a perfect square at all.)

7. Let m be a fixed positive integer.

(a) Prove that no two among the integers 0, 1, 2, . . . , m − 1 are congruent to each other modulo m.

(b) Prove that every integer is congruent modulo m to one of 0, 1, 2, . . . , m − 1.