One method of slowing the growth of an insect population without using pesticides is to introduce into the population a number of sterile males that mate with fertile females but produce no offspring. Let P represent the number of female insects in a population and S the number of sterile males introduced each generation. Let r be the per capita rate of production of females by females, provided their chosen mate is not sterile. Then the female population is related to time t by

t =

Suppose an insect population with 10,000 females grows at a rate of

r = 1.8

and 700 sterile males are added. Evaluate the integral to give an equation relating the female population to time. (Note that the resulting equation can't be solved explicitly for P. Remember to use absolute values where appropriate.)

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.