Find the eigenfunctions for the following boundary value problem.

x²y?? ? 15xy? + (64 + ?)?y ?=? 0,    y(e^?1) ?=? 0, ?y(1) ?=? 0.

In the eigenfunction take the arbitrary constant (either c₁ or c₂) from the general solution to be 1

A circle is divided in 6 sectors by 3 diameters.
Each sector contains a pawn. We are allowed to chose two pawns and
move each of them to a sector bordering the one it stands on at the
moment.
Is it possible to gather all 6 pawns in one sector using such
operations? Prove your answer

1.Find the present value at time 0 of $15086 due at the end of 4.88 years if the force of interest δ=0.023δ=0.023.

2.If an investment will double in 8.15 years at a constant force of interest δ, then

3.An investment of $1300 at t = 0 accumulates at a constant force of interest δδ= 4% for the first 4 years and at a nominal annual rate of interest of 5% compounded semiannually thereafter. Find the accumulated value of this investment at time t = 11.

4.An investment pays $1150 at time 0 and $2250 at the end of 3 years. Find the accumulated value of this investment at time 8 if the force of interest δt=0.02(1+t)2δt=0.02(1+t)2.

5.An investment of $1700 at t = 4 accumulates at a force of interest δt=0.003+0.009t2δt=0.003+0.009t2. Find the accumulated value of this investment at time t = 9.

6.How long does it take an amount to triple if the force of interest δ=0.062δ=0.062.

if a in G (group ) such as o(a)=mn

prove the existence of g and h in G such as a=gh=hg and o(g)=m o(h)=n

Historically speaking, several attempts have been made to create ‘metric time’ using factors of 10, but our current system won out. If 1 day was 10 metric hours, 1 metric hour was 10 metric minutes, and 1 metric minute was 10 metric seconds, what time would it really be if a metric clock reads 2:9:1? Similarly, convert 11:13:31 P.M. to metric time. You may assume that each new day starts at midnight.

The definition of a (well-defined) function f : X → Y . Meaning of domain, range, and co-domain

Let S be the two dimensional subspace of R^4 spanned by
x = (1,0,2,1) and y = (0,1,- 2,0)
Find a basis for S^⊥

Customers arrive in a certain shop according to an approximate Poisson process on the average of two every 6 minutes.

(a) Using the Poisson distribution calculate the probability of two or more customers arrive in a 2-minute period.

(b) Consider X denote number of customers and X follows binomial distribution with parameters n= 100. Using the binomial distribution calculate the probability oftwo or more customers arrive in a 2-minute period.

(c) Let Y denote the waiting time in minutes until the first customer arrives. (i) What is the pdf ofY? (ii) Find q1=π0.75

(d) Let Y denote the waiting time in minutes until the first customer arrives. What is the probability that the shopkeeper will have to wait more than 3 minutes for the arrival of the first customer ?

(e) What is the probability that shopkeeper will wait more than 3 minutes before both of the first two customers arrive?

Let A be an m x n matrix and b and x be vectors such that Ab=x.

a) What vector space is x in?

b) What vector space is b in?

c) Show that x is a linear combination of the columns of A.

d) Let x' be a linear combination of the columns of A. Show that there is a vector b' so that Ab' = x'.

Do the polynomials x^3 + x + 2, x^2 - x +2, -x^3 + x^2 - 5x + 3, and x^3 + 2 span P3? (show your conclusion.)

Prove that if G is a connected graph of order n is greater than or equal to 3, then its square G^(2) is 2-connected

5. (a) Prove that the set of all real numbers R is uncountable.

(b) What is the length of the Cantor set? Verify your answer.

1.

A solid in the first octant, bounded by the coordinate planes, the plane (x= 40) and the curve (z=1-y² ) , Find the volume of the solid by using : a- Double integration technique ( Use order dy dx) b-Triple integration technique ( Use order dz dy dx)

2.

Use triple integration in Cartesian coordinates to find the volume of the solid that lies below the surface = 16 − ?² − ?² , above the plane z = 40/ 100 , and bounded by the curve ? = √? and the lines ? = ? − 2 and ? = −?.

3.

Find all the local maxima, local minima, and saddle points of the function ?(?, ?) = 40?² − 2?³ + 3?² + 6?y

4.

Let ? = ? − sin(??) ?ℎ??? ? = 40? ? = ln(?) ? = e^(t-1)Find ??/?? by :- a- Using Chain Rule principles
b- Expressing (?) in term of (?) then differentiating directly

5.

a- Use implicit differentiation to find (d²y)/(dx² ) of the following curve at the point (π, 2π).y² = x²+ sin ?y

b- Show that ?⁵?⧸ ??²??³ is zero in two differentiation steps only. ?(?, ?) = ??^ ?²/⁴⁰
محمد مهدي جميل ?, [⁧يونيو ⁨30⁩، ⁨2020⁩ في ⁨18:22⁩⁩]
Show that ?⁵?/ ??²??³ is zero in three differentiation steps only. ?(?, ?) = ?² + ?(sin ? – ?⁴ ).