Solve differential equation by using undetermined coefficient method

y^''+9y=x^2e^x+6 , y(0)=1 and y^'(0)=1

Four numbers have a sum of 9900. The second exceeds the first by 1/7 of the first. The third exceeds the sum of the first two by 300. The fourth exceeds the sum of the first three by 300. Find the four numbers.

The polynomial equation x^3−5x^2+11x−15=0 is known to have an integer solution. Complete the following table listing in the first column the candidate integer solutions (there are eight) of x^3−5x^2+11x−15=0 supplied by the Rational Root Test, and in the second column the values of P evaluated at the corresponding candidates. MAKE SURE THAT THE CANDIDATE ROOTS ARE IN INCREASING ORDER!

x. x^3−5x^2+11x−15

#1

#2

#3

#4

#5

#6

#7

#8

With this information, give all three roots of PP (with distinct roots separated by a comma):

Q1)

(a) Write implicit equations for two parallel planes, P1 containing the line <−3,−5,3>+t<3,−4,−5> and P2 containing the line <−1,−2,1>+t<5,4,1>

(b)  Let n = <16,-28,32>

The set of vectors w = <x,y,z> such that <3,−4,−5> x w = n forms a line. Write a parametric equation for that line, and make sure to use t as your parameter.

Please show the working clearly.

solve the following initial value problem y''+4y'=g(t),y(0)=0,y' (0)=1 if g(t) is the function which is 1 on [0,1) and zero elsewhere

Laplace Transform : y ' - y = e^-3t cos3t , y(0) =3

and, Show that, Differential Form ?

dU = Tds - Pdv , dH=Tds-Vdp , dF= -sdT-Pdv , dG= -sdT+VdP

Find the second-order Taylor polynomial for f(x,y)=8y^2e^(−x^2) at (1,1).

(Use symbolic notation and fractions where needed.)

p(x,y)p(x,y) = .

Give examples of sets of three vectors that are:

a) Collinear

b) Coplanar

c) Not coplanar

Find the general power series solution to the following ode:

y"-4xy'+(4x²-2)y=0

Consider the following LOP P.
Max. z = 212x1 −320x2 +273x3 −347x4 +295x5
s.t. −4x1 −2x3 +8x5 ≤ −22
2x1 +3x2 −x4 = 31
−5x2 +3x3 −2x5 ≤ 27
−7x1 −8x3 +6x4 = −38
−9x3 −2x4 +x5 ≤ −40
−x2 −3x4 −5x5 ≤ 42
& x1, x3, x4 ≥ 0
a. Find x∗ and write the Phase 0, I and II pivots that solve P.
b. Use the General Complementary Slackness Theorem to find
the optimal certificate y∗
[do not solve the dual LOP D!].

z² = xy-3x+9 Find the point closest to the origin on the surface.

The equation of parabola is x2 = 4ay . Find the length of the latus rectum of the parabola and length of the parabolic arc intercepted by the latus rectum. (a) Length of latusrectum : 4a; Length of the intercepted arc : 4.29a (b)Length of latusrectum: —4a; Length of the intercepted arc : 4.39a (c) Length of latusrectum: —4a; Length of the intercepted arc : 4.49a (d) Length of latusrectum: 4a; Length of the intercepted arc : 4.59a

A utility-maximizing consumer who consumes quantity x and y of two goods has a quadratic utility function given by Ux,y=x+y-0.05x2-0.05y2 subject to

2x+5y=128

Where $128 is the consumer’s budget and the prices of the two goods are, respectively, 2 and 5. Assuming marginal utilities Ux, Uy > 0,

a. Find the quantity x and y that maximize the utility function.

b. Using bordered Hessian, check utility for a maximum.

c. What is the maximum utility of the consumer?

Benoit believes that he didn’t invent the Mandelbrot set. Are complex numbers invented or discovered? Whichever position you take, please research from reputable sources how complex numbers came about and make sure to include views that support both sides of the argument in your response. (Remember to cite your sources. (Minimum two paragraphs, five complete sentences each.)