use laplace to answer y"-3y'+2y=1+cost+e^-t,y(0)=1,y'(0)=0

Consider a sample with data values of 27, 25, 20, 15, 30, 34, 28, and 25. Compute the 19th, 24th, 60th, and 70th percentiles.

If needed, round your answers to two decimal digits.

Percentile 19% 24% 60% 70%

Value for percentile?

(a) Find the matrix representation for the orthogonal projection Pr : R 4 → R 4 onto the plane P= span

(b) Find the distance of vector ~y =

from the plane P.

Solve each measuring problem, or explain why it can’t be done. (You have unlimited water.)

Using 6 and 15 gallon jugs, measure

(i) 3 gallons (ii) 4 gallons (iii) 5 gallons

Explain the Riemann series theorem proof.

Prove the following statement:

Suppose it's your turn and the Nim sum of the number of coins in the heaps is equal to 0. Then whatever you do, the Nim sum of the number of coins after your move will not be equal to 0.

Let R and S be nontrivial rings (i.e., containing more than just the 0 element), and define the projection homomorphism pi_1: R X S --> R by pi_1(x,y)=x.

(a) Prove that pi_1 is a surjective homomorphism of rings.

(b) Prove that pi_1 is not injective

(c) Prove that (R X S)/R (not congruent) R.

Please show all parts of answer

Let {xn} be a real summable sequence with xn ≥ 0 eventually. Prove that √(Xn*Xn+1) is summable.

That is, PDEs of the general form A(x, y, u) ∂u(x, y) ∂x + B(x, y, u) ∂u(x, y) ∂y = C(x, y, u), (1) for some A, B and C. To solve such PDEs we first find characteristics, curves in the solution space (x, y, u) parametrically given by (x(τ ), y(τ ), u(τ )), which satisfy dx dτ = A(x, y, u), dy dτ = b(x, y, u), du dτ = C(x, y, u). (2) We find solutions to these equations in the form f(x, y, u) = C1 and g(x, y, u) = C2 where C1 and C2 are arbitrary constants. The independent functions f and g are then used to write the general solution to Equation (1) f(x, y, u(x, y)) = F [g(x, y, u(x, y))] , (3) where F is a sufficiently smooth function (that is, you can expect in this question that its derivative exists everywhere). 1. [12 marks] Verify that (3) for any suitable F and for any f and g as described above is actually a solution to the PDE (1). That is, you should show that given (2) which describe the functions f and g and the solution (3), then Equation (1) is always satisfied. HINT: This is not as simple as it sounds. You should first attempt to differentiate f(x, y, u) = C1 and g(x, y, u) = C2 by τ and differentiate the solution (3) first with respect to x and then with respect to y and use the resultant simultaneous equations to deduce (1). 1 2. [11 marks] Show that the implicit equation u = y + F x + y u − log (u) for sufficiently smooth arbitrary F is a solution to the PDE y ∂u ∂x + u 2 ∂u ∂y = u 2 .

Can u kindly do both of them, or just do question 1,thanks

In lectures, we have discussed first order quasilinear PDEs. That is, PDEs of the general form A(x, y, u) ∂u(x, y) ∂x + B(x, y, u) ∂u(x, y) ∂y = C(x, y, u), (1) for some A, B and C. To solve such PDEs we first find characteristics, curves in the solution space (x, y, u) parametrically given by (x(τ ), y(τ ), u(τ )), which satisfy dx dτ = A(x, y, u), dy dτ = b(x, y, u), du dτ = C(x, y, u). (2) We find solutions to these equations in the form f(x, y, u) = C1 and g(x, y, u) = C2 where C1 and C2 are arbitrary constants. The independent functions f and g are then used to write the general solution to Equation (1) f(x, y, u(x, y)) = F [g(x, y, u(x, y))] , (3) where F is a sufficiently smooth function (that is, you can expect in this question that its derivative exists everywhere). 1. [12 marks] Verify that (3) for any suitable F and for any f and g as described above is actually a solution to the PDE (1). That is, you should show that given (2) which describe the functions f and g and the solution (3), then Equation (1) is always satisfied. HINT: This is not as simple as it sounds. You should first attempt to differentiate f(x, y, u) = C1 and g(x, y, u) = C2 by τ and differentiate the solution (3) first with respect to x and then with respect to y and use the resultant simultaneous equations to deduce (1).

Let K be a cone with a circular bottom, that has a radius r, and the apex is directly above the center of the bottom. Let h represent the height of the cone. Show that the surface area of the cone K without the bottom is equal to

pi * r * sqrt(r^2 + h^2) .

(Use that a sector that is given with angle θ in a circle with radius R has the area  (θ * R^2)/2

Suppose that a large mixing tank initially holds 400 gallons of water in which 65 pounds of salt have been dissolved. Another brine solution is pumped into the tank at a rate of 6 gal/min, and when the solution is well stirred, it is pumped out at a slower rate of 5 gal/min. If the concentration of the solution entering is 3 lb/gal, find the amount of salt in the tank after 10 minutes.

PLEASE SHOW ALL WORK AND USE DIFF. EQs. THANKS!