Solve the following differential equation. "Preferably using the chain rule/ order reduction".

(d) y′y′′−3(y′)^{2}=y^{2} y′′.

(e) (y′)2y′′=y.

Postal regulations specify that a parcel sent by parcel post may have a combined length and girth of no more than 130 in. Find the dimensions of a cylindrical package of greatest volume that may be sent through the mail, using the method of Lagrange multipliers

4. Let n ≥ 8 be an even integer and let k be an integer with 2 ≤ k ≤ n/2. Consider k-element subsets of the set S = {1, 2, . . . , n}. How many such subsets contain at least two even numbers?

Consider the nonlinear equation f(x) = x³− 2x² − x + 2 = 0.

(a) Verify that x = 1 is a solution.

(b) Convert f(x) = 0 to a fixed point equation g(x) = x where this is not the fixed point iteration implied by Newton’s method, and verify that x = 1 is a fixed point of g(x) = x.

(c) Convert f(x) = 0 to the fixed point iteration implied by Newton’s method and again verify that x = 1 is a fixed point.

(d) Write MATLAB code to iterate on your fixed point iteration as well as the fixed point iteration implied by Newton. Compare these results based on x0 = 1.1. How fast did Newton converge? How fast did your iteration from part b converge (or did it at all)? What does the theory of fixed points tell you about these convergence results?

Let N(n) be the number of all partitions of [n] with no singleton blocks. And let A(n) be the number of all partitions of [n] with at least one singleton block. Prove that for all n ≥ 1, N(n+1) = A(n). Hint: try to give (even an informal) bijective argument.

an LTIC system is specified by the equation (D² +4D + 4)y(t) = Dx(t) with initial condition y'(0) = -4... How do I find the other initial condition?

Write down the chromatic polynomials of
(i)the complete graph K7;
(ii)the complete bipartite graph K1,6.
In how many ways can these graphs be coloured with ten colours?

Find the solution of the system

x′=−4y, y′=3x,

where primes indicate derivatives with respect to t, that satisfies the initial condition x(0)=1, y(0)=−1.

4. Prove that for any n∈Z+, An ≤Sn.

find the appropriate series solution or solutions, using the frobenius method, about the origin (x_0 = 0)

2x(x+2)y'' + y' -xy = 0

Show by induction that if a prime p divides a product of n numbers, then it divides at least one of the numbers.

Number theory course. Please, I want a clear and neat and readable answer.

5. Another equation that has been used to model population growth is the Gompertz equation dP /dt = rP ln (K P ) , where r and K are positive constants, and P(t) > 0.

(a) Sketch a clearly labeled graph of f(P), where dP/dt = f(P) (state the main facts you used to obtain your answer).

(b) Draw the phaseline. Next to it, sketch a set of integral curves in the t − P plane.

(c) State and classify all equilibria.

(d) (Extra Credit) For 0 < P < K, find P 00(t) and determine where the graph of P(t) is concave up and where it is concave down. Enter the inflection point in your graph in (a). (Enter your answer in an attached blank page.)