A​ third-order homogeneous linear equation and three linearly independent solutions are given below. Find a particular solution satisfying the given initial conditions.

y⁽³⁾+2y''-y'-2y=0; y(0)=1, y'(0)=2, y''(0)=0,

y_{1=e^x ,} y₂=e^-x, y₃=e^-2x

Find the particular solution.

Let L be the line parametrically by~r(t) = [1 + 2t,4 +t,2 + 3t] and M be the line through the points P= (−5,2,−3) and Q=(1,2,−6).

a) The lines L and M intersect; find the point of intersection.

b) How many planes contain both lines?

c) Give a parametric equation for a plane Π that contains both lines

prove that every integer greater than 2 can be broken into sum of multiples of 2 and 3

A tank initially contains 80gal of pure water. Brine containing 2lb of salt per gallon enters the tank at 2gal/min, and the (perfectly mixed) solution leaves the tank at 3 gal/min. Thus the tank is empty after exactly 80 min.

(A) Find the amount of salt in the tank after t minutes.

(b) What is the maximum amount of salt ever in the tank?

Let F be a vector field. Find the flux of F through the given surface. Assume the surface S is oriented upward. F = eyi + exj + 24yk; S that portion of the plane x + y + z = 6 in the first octant.

1. Let v1, . . . , vn be nonzero vectors such that each vi+1 has more leading 0s than vi . Show that vectors v1, . . . , vn are linearly independent.

Consider the solution of the differential equation y′=−3yy′=−3y passing through y(0)=0.5y(0)=0.5.

A. Sketch the slope field for this differential equation, and sketch the solution passing through the point (0,0.5).

B. Use Euler's method with step size Δx=0.2Δx=0.2 to estimate the solution at x=0.2,0.4,…,1x=0.2,0.4,…,1, using these to fill in the following table. (Be sure not to round your answers at each step!)

C. Plot your estimated solution on your slope field. Compare the solution and the slope field. Is the estimated solution an over or under estimate for the actual solution?
A. over
B. under

D. Check that y=0.5e−3xy=0.5e−3x is a solution to y′=−3yy′=−3y with y(0)=0.5y(0)=0.5.

Calculate the first three terms in the power series solutions of the following differential equations taken about x=0.

x^2y''+sinx y'-cosx y=0

Please solve the recurrence relation by finding the explicit formula of each problem. Show all work, thanks!

A. c_k = 6c_k-1 - 9c_k-2      k≥2,   c₀ =1, c₁=3

B. d_k = 2d_k-1 +k                       k≥1,   d₀ =1, d₁=3

C. a_k = 3a_k-1 + 2                         k≥1,   a₀ =3

D. b_k = -b_k-1 + 7b_k-2    k≥2,   b₀ =1, b₁=4

Prove Mn(reals) is a group under matrix addition. Note, Mn(reals={A|A is a real nxn matrix}) Please show all steps and do not write in script!

Consider equation: mx”+cx’+kx=F0cos(\omega t) Find the transient motion and steady periodic oscillations of a damped mass and spring system with m=1; c=2; and k=26 under the influence of an external force F(t)=82cos(4t) with x(0)=6 and x’(0)=0.

Exercise 1. Write an algorithm (pseudocode) to read a set of sales data items from standard input and calculate and output their total and their average. Prompt user to enter number of data items. Exercise 2. Create a test data set to verify your algorithm. How many cases are needed? Explain. Write your test data set below for submission to the EOL dropbox upon completion of your lab. Number of items List data items Expected output Case 1: Exercise 3. Create a flowchart for your algorithm on Raptor and verify it using your test data. Copy and paste your flowchart below for submission to the EOL dropbox upon completion of your lab. Exercise 4. Write a C++ program that implements your flowchart. Use a do-while loop for input validation and a for loop to calculate the total (You may skip this exercise until we cover for and do-while loops)