if L1(x) = x is a solution to the ODE:

(1-x)L'' - 2xL' +2L = 0

a.) Show that L1(x) = x is a solution to the ODE

2.) Use reduction of order to find another solution L2(x)

Find the local maximum and minimum values and saddle point(s) of the function. If you have three-dimensional graphing software, graph the function with a domain and viewpoint that reveal all the important aspects of the function. (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.)

f(x, y) = x³ + y³ − 3x² − 3y² − 9x

Solve y''-y=e^t(2cos(t)-sin(t)) with initial condition y(0)=y'(0)=0

Use induction to prove

Let f(x) be a polynomial of degree n in Pn(R). Prove that for any g(x)∈Pn(R) there exist scalars c0, c1, ...., cn such that g(x)=c0f(x)+c1f′(x)+c2f′′(x)+⋯+cnf(n)(x), where f(n)(x)denotes the nth derivative of f(x).

Solve the initial value problem Utt = c^2Uxx,

u(0,x)= e^-x^2 , Ut(0,x) = sin x

(PDE)

Problem 13.6. (a) How many different relations are on X = {1, 2, 3}.
(b) How many different equivalence relations are on this set?

Two people are on a city block. Person A is on the northeast corner and Person B is on the southwest corner. Person A starts walking towards the southeast corner at a rate of 3 ft/sec. Four seconds later Person B starts walking towards the southeast corner at a rate of 2 ft/sec. At what rate is the distance between them changing (a) 10 seconds after Person A starts walking and (b) after Person A has covered half the distance?

Suppose u(t,x)solves the initial value problem Utt = 4Uxx + sin(wt) cos(x),

u(0,x)= 0 , Ut(0,x) = 0. Is h(t) = u(t,0) a periodic function?

(PDE)

True or False and support your decision

1.) if Ax=0 has only one trivial solution, then Ax=0 has a unique solution.

2.)The linear system given by Ax=0 consisting of 3 equations and 4 unknowns has infinitely many solutions

3.) If the vector set {v,u,w} is linearly independent, then w is a linear combination of u and v

4.) If A is a 3by5 matrix with 3 pivots,then the columns of A span R5

Find the smallest n ∈ N such that 2(n + 5)^2 < n^3 and call it n^0，Show that 2(n + 5)^2 < n^3 for all n ≥ n^0.

Find at least the first four nonzero terms in a powerseries expansion about x = 0 for a general solution to thegiven differential equation: Include a general formula for the coefficients (recurrence formula). x=0; (x^2 +4)y'' + y=x

Let τ (n) denote the number of positive divisors of n and σ(n) denote the sum of the positive divisors of n (as in the notes).

(a) Evaluate τ (1500) and σ(8!).

(b) Verify that τ (n) = τ (n + 1) = τ (n + 2) = τ (n + 3) holds for n = 3655 and 4503.

(c) When n = 14, n = 206 and n = 957, show that σ(n) = σ(n + 1).