### Solve the equation

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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 <

Please see attachment

Please see attachment

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

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.

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|

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

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

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

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

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

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

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. 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.

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

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.

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.

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).

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