A mass-spring-dashpot system is described by

my′′ + cy′ + ky = F_o cos ωt,

see §3.6 Eq. (17). This second-order differential equation has been used in simulations, such as this

one at the PhET site: https://phet.colorado.edu/en/simulation/legacy/resonance.

For m = 2.53kg, c = 0.502N/(m/s), k = 97.2N/m, F_o = 97.2×0.5N = 48.6N,and ω = 2.6, the equation becomes

2.53y′′ + 0.502y′ + 97.2y = 48.6 cos(ωt) ....................... (1)

(a) Given the initial value

y(0) = 2.20, y′(0) = 0,

solve Eq. (1), Round to three significant figures. Show all your work, and clearly highlight your

conclusion—the combination of complementary function and particular solution.

(b) From your particular solution, which is of the form

y_p =Acosωt+Bsinωt,

calculate the amplitude, which is √(A2 + B2). Clearly highlight your conclusion—the amplitude. You are encouraged to use the PhET simulation to verify your amplitude. Note that the angular frequency ω = 2πf where f is the frequency in the simulation.

If (x,y,z) is a primitive Pythagorean triple, prove that z= 4k+1

What is the general solution to xy''+y'+k^2xy=0

How do you solve 2xy''+y'+x^3y=0 Using the forbenius method

One can encrypt numbers as strings of letters by applying a substitution cipher.

For example, the substitution 0 <-> F, 1 <-> G, 2 <-> D, 3 <-> Z, would encrypt the number 30231 as the string of letters ZFDZG.

Decipher the following encrypted equation involving three numbers: AB + BC + ACA = BCB.

(You may need to use the fact that our usual numbers are represented in base 10, for example, the number 838 is represented with digits 8, 3, 8, since 838 = 8 * (100) + 3 * (10) + 8.

Using the epsilon-delta definition prove

Lim of x^2 =4 when x approaches 2

Problem 1: Consider the following Initial Value Problem (IVP) where ? is the dependent variable and ? is the independent variable: ?′=sin(?)∗(1−?) with ?(0)=?0 and ?≥0
Note: the analytic solution for this IVP is: ?(?)=1+(?_0−1)?^cos(?)−1

Part 1A: Approximate the solution to the IVP using Euler’s method with the following conditions: Initial condition ?_0=−1/2; time step ℎ=1/16; and time interval ?∈[0,20]
+ Derive the recursive formula for Euler’s method applied to this IVP
+ Plot the Euler’s method approximation
+ Plot the absolute error between the approximation and the exact solution using a semilog plot

Coding errors, please type out.

Let W be an inner product space and v1, . . . , vn a basis of V . Show that
<S, T> = <Sv1, T v1> + . . . + <Svn, T vn> for S, T ∈ L(V, W) is an inner product on L(V, W).

1. Find the solution of y" + 8y' = 896sin(8t) + 256cos(8t)

with y(0) = 9 and y'(0) = 9

y = ?

2. Find y as a function of x if y"' - 9y" + 18y' = 20e^x ,

y(0) = 30 , y'(0) = 23 , y"(0) = 17

y(x) = ?

Find y as a function of x if

y''''−4y'''+4y''=−128e^{-2x}

y(0)=2,  y′(0)=9,  y″(0)=−4,  y‴(0)=16.
y(x)=?

in parts a and b use gaussian elimination to solve the system of linear equations. show all algebraic steps.

a. x1 + x2 + x3 = 2

x1 - x3 = -2

2x2 + x3 = -1

b. x1 + x2 + x3 = 3

3x1 + 4x2 + 2x3 = 4

4x1 + 5x2 + 3x3 = 7

2x1 + 3x2 + x3 = 1

Let H, K be groups and α : K → Aut(H) be a homomorphism of groups. Show that H oα K is the internal semidirect product of subgroups which are isomorphic to H and K, respectively

A spring with a 6-kg mass and a damping constant 5 can be stretched 1.5 meters beyond its natural length and held at rest there by a force of 7.5 newtons. Suppose the spring is stretched 3 meters below spring-mass equilibrium and then released with zero velocity.

Find the position of the mass after t seconds.

y(t)=?

Please show all the work.

a) Show that 6, 28, 496, 8128, and 33550336 are perfect numbers (recall, according to the note: n is said to be perfect if σ(n) = 2n).

b) Recall that prime numbers of the form M_n := 2ⁿ − 1 are called the Mersenne primes. For those nsuch that M_n := 2ⁿ − 1 is prime,

prove that the number P_n := 1/2 (M_n + 1)M_n= 2^(n-1)(2ⁿ − 1) is a perfect number (Note: for P₁ = 6, P₂ = 28, P₃ = 496, P₄ = 8128, P₅ = 33550336 which recover the perfect numbers in (a)).

c) Let P = q · 2^(n-1) where q is an odd prime. Prove that if P is a perfect number, then q = 2ⁿ − 1, i.e. all perfect number of the form P = q · 2^(n-1) is of the form 2^(n-1) (2ⁿ − 1).