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# Calculus and Analysis

### Equation for Population P

View attachment for the problem.

### Use the Uniqueness Theorem for the Initial Value Problem

4. For the initial value problem dy/dx = 3y^(2/3), y(2) = 0, (a) does existence uniqueness Theorem 1 imply the existence of a unique solution? Explain. (b) Which of the following functions are solutions to the above differential equation? Explain. (b_1) y(x) = 0 (b_2) y(x) = (x - 2)^3 (b_3) y(x) = (x - alpha)^3, x <

### How do you find the inverse laplace transform

Find the inverse laplace transform: (L^-1) * [1 / (s^2 * (s^2 + 1))]

### How do you find the inverse laplace transform?

Find the inverse Laplace transform: (L-1) * [(e^-2s) / (s^2)]

### Inverse Laplace Transform

Please see the attached file for the fully formatted problems.

### Derivative Question - Algebraic Function

This is from a Trig/Calculus course...Explain FULLY: If F(x) = x^4 - 2x^3 + 4x^2 - 9 Note: ^ indicates exponant. Find F prime of x. It will be a derivative. I need every step explained clearly as I have a bet riding on this! I need to be able to show every step in order to win my bet.

### Differential Equation : General Solution

Please see the attached file for the fully formatted problems. Find a general solution on (-pi/2,pi/2) to y''+y=tan x given that S sec x dx = ln|secx + tanx|

### Differential Equations : Method of Undetermined Coefficients

Use the method of undetermined coefficients to solve the following differential equation. y'' + 2y' - 3y = 9x - 10 sin x y(0)=0 y'(0)=4

### Differential Equations : Method of Undetermined Coefficients

Decide whether the method of undertermined coefficients can be applied to find a particular solutions of the given equations. (Explain) a) y'' + 3y' - y = tan x b) y'' + xy' + y = sin x

### Find the solution (y sub p) to the following differential equation.

Y'' + 2y' - 3y = e^(-3x) + x^2 * e^x

### Differential Equation : Homogeneous Solution

Y'' + p(x)y' + q(x)y = r(x) has three solutions sin x, cos x, and sin 2x. Find yh. (yh is the corresponding homogeneous solution)

### Differential Equation : Wronskian

Y'' + p(x)y' + q(x)y = 0 has two solutions x^2 - x and x^3 - x. Use the Wronskian to find p(x).

### Differential Equation Functions

Y'''' - 2y''' + 2y'' - 2y' + y = 0

### Differential Equation : Find a General Solution

3x^2y'' + 11xy' - 3y = 0, x>0

### Determine whether the following functions can be Wronskians on -1<x<1 for a pair of solutions to some equation y''+py'+qy=0 with p and q continuous.

Determine whether the following functions can be Wronskians on -1<x<1 for a pair of solutions to some equation y''+py'+qy = 0 with p and q continuous. a) W(x) = 6e^4x b) W(x) = x^3 c) W(x) = 0 d) W(x) = (x-1/2)^2

### Equations: Linear or Nonlinear

A) yy''-y' = sin x b) x^2y''-y'+y = cos x.

### Assorted Differentiation and Tangent to Curve Problems

A. i) Differentiate the equations given as items 21 and 22 on your worksheet. ii) Refer to the formula given as item 23 of your worksheet. The equation relates to one particular machine in an engineering workshop. The machine sots C pounds to lease each week according to the formula and 't' is the number of hours per wee

### Laplace Transform Differential Equations

If L[f(t)]=F(s) then L[t*f(t)]= -dF/ds use this result to compute L[t*e^kt].

### Laplace Transform Solutions

Use the laplace transform to solve the ODE y"+3y = cos(2t), y(0)=0 , y'(0)=0 Show all details related to using the inverse transform.

### Horse Velocity Using Derivatives

The problem is in JPEG, thank you. Quarter horses race a distance of 440 yards (a quarter mile) in a straight line. During a race the following observations where made. The top line gives the time in seconds since the race began and the bottom line gives the distance (in yards) the horse has traveled from the starting line.

### Population growth differential equation.

The birth rate in a state is 2% per year and the rate is 1.3% per year. The population of the state is now 8,000,000. a) At what rate are babies being born in the state now? with units b) At what rate are people dying in the state now? c) Write a differential equation that the population of the state satisfies. include

### Application of Stokes Theorem

Use Stokes' Theorem to evaluate int (F.dr) over C where F = x^2*y i +x/3 j +xy k and C is the curve of intersection of hyperbolic paraboloid z= y^2-x^2 and teh cylinder x^2+y^2=1 oriented counterclockwise.

### Vector Field Sketches and Flow Lines

Sketch the vector fields and flow lines (See #34 Attached for full question)

### The Existence Theorem for Nonlinear Differential Equations

Please see the attached file for the fully formatted problems. Let g(x,y) be Lipschitz continuous. Let ? (x) = y , and for n > 0 define ? (x) = y + Prove that ? (x) &#61614; ?(x) on [x - , x + ], for some > 0, where ?(x) solves the ODE ?'(x) = g(x, ?(x)), and ?(x ) = y

### The Mean Value Theorem and Directional Derivatives

Please see the attached file for the fully formatted problems. Let F: R^n --> R be continuously differentiable. Show that at each point x E R^n there is a direction hx so that the directional derivative is 0, i.e., df/dhx (x) = 0. Is hx unique? Give a method for determining hx.

### Second-Order Approximation and the Second-Derivative Test

1. Let f(x,y) = xy + 2x y - 6xy (a) Locate the critical points of f(x,y) and determine if they are local maxima, minima, or neither. (b) Find the first and second order approximations of f(x,y) at the point (1,-1).

### Implicit Function Theorem

Please see attachment. Thank you. Use the Implicit Function Theorems to show that the system of equations:

### Simulation : Skydiver in Free-fall

A skydiver, weighing 70kg, jumps from an aeroplane at an altitude of 700 metres and falls for (T) seconds before pulling the rip cord of his parachute. A landing is said to gentle if the velocity on impact is no more than the impact velocity of an object dropped from a height of 6 metres. The distance that the skydiver falls d