### Inverse Laplace Transform

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

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

Please see the attached file for the fully formatted problems.

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.

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

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

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)

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).

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

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

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

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

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.

The problem is in JPEG, thank you.

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

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

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)  ?(x) on [x - , x + ], for some > 0, where ?(x) solves the ODE ?'(x) = g(x, ?(x)), and ?(x ) = y

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.

Please see attachment. Thank you.

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

Use Lagrange multipliers to determine the smallest value of the function f(x,y) = 2x^2-x+3y^2 for points (x,y) on the circle x^2 + y^2 = 1.

I can't figure out exactly how to formulate a riemann sum. For example, when given y=x+2; [0,1], and told to "find the area of the region under the curve y=f(x) over the interval [a,b]. To do this, divide [a,b] into n equal subintervals, caluculate the area of the cooresponding circumscribed polygon, and then let n go to infin

34) The function f is continuous on the closed interval [1,5] and has values that are given in the table below. If 2 subintervals of equal length are used, what is the midpoint Reimann sum approximation of integral with 5 on top and 1 on bottom f(x)dx? Please given step by step explaination and answer is 32. x 1 2 3

36)If the functions f and g are defines for all real numbers and f is an antiderivative of g, which statements are not true? I If g(x)>0 for all x, then f is increasing. II If g(z)=0 then f(x) has a horizontal tangent at x+a. III If f(x)=0 for all x, then g(x)=0 for all x. IV If g(x)=0 for all x, then f(x)=0 for all x.

24) Let f be a differentibale function defined on the closed interval [a,b] and let c be a point in the open interval (a,b) such that. I f'(c)=0 II f'(x)>0 when a<or equal to x<c, and III f'(x)<0 when c<b<or equal to b. Which is true? Then tell why others false. a. F'(c)=0 b. F"(c)=0 c. F(c) is an abs. ma

26) The verticle height in feet of a ball thrown upward from a cliff is given by s(t)=-16t^2+64t+200, where t is measured in seconds. What is the height of the ball, in feet, when its velocity is zero? 27)If the function f is continuous for all real numbers and the lim as h approaches 0 of f(a+h)-f(a)/h = 7 then which state

Using P@t=P@initial time(1+r)to the t power. where r is the percentage rate increase and t is the difference in ending year and beginning year. Find equation for P @any t. Use this chart: Year 1825 1850 1875 1900 1925 1950 1975 P(in 100 152 218 301 404 535 700 thousands)

A) y(x) = 0 b) y(x)= (x-2)^3 c) y(x) = {(x-a)^3, x<a {0, a<x<b {(x-b)^3, x>b where a<2<b

B) Explain why the Euler's method cannot be used to approximate y(2)?

Please see the attached file for the fully formatted problem. Find the critical points and use your test of choice to give local maximum and minimum values. Give those values. f(x) = x^2 /sqrt(x^2 + 4)

Please see the attached file.