64. (The clepsydra, or water clock) A 12 hour water clock is to be designed with the dimensions shaped like the surface obtained by revolving the curve y = f(x) around the y-axis. What should be this curve, and what should be the radius of the circular bottom hole, in order that the water level will fall at the constant rate of 4 inches per hour?

Height = 4 ft
Radius of the top of the clock = 1 ft

Hint: A(y) dy/dt = - k* sqrt(y)

43. Arthur Clarke's The Wind from the Sun describes Diana, a spacecraft propelled by the solar wind. Its aluminized sail provides it with a constant acceleration of 0.0001 g = 0.0098 m/s2. Suppose this spacecraft starts from rest at time t = 0 and simultaneously fires a projectile (straight ahead in the same direction) that travels at one-tenth of the speed c = 3 x 108 m/s of light. How long will it take the spacecraft to catch up with the projectile, and how far will it have traveled by then?

Solution Summary

The solution shows how to set up and solve the differential equation in two practical scenarios.

I have this differentialequation.
y' = x^2*cos^2(y).
I don't know if it's a separable differentialequation or a first order differentialequation. Please show me the steps in how to solve it.
Thank you.

1. Complex Exponentials: Simply the following expression and give your answer both in polar and rectangular form.
o c=3ejπ/4+4e−jπ/2
2. Difference Equations: Solve the following difference equation using recursion by hand (for n=0 to n=4)
o y[n] + 0.5y[n-1] = 2x[n-1]; x[n] = δ[n], y[-1] = 0
3. DifferentialEquations

Find the general solution of the second order differentialequation y'' - y' - 6y = e^-3x
This one is quite long winded, and I am pretty sure that I am getting yh right but can't seem to get close to yp. I think this is a D.R.A.E?

Dy/dx= 4x+ 9x^2/(3x^3+1)^3/2
given point: (0,2)
(that's 4x plus 9x squared over (3x cubed + 1) to the 3/2 power)
Use the differentialequation and the given point to find an equation of the function.

Use D-operators to find a particular solution to the differentialequation:
y^n + y' - 2y= e^-2x
Hence write its general solution. Find the solution that satisfies the initial conditions:
y(0) = 1/3, y'(0) = -1/3

(b) Find a suitable integrating factor R for the differentialequation and verify whether the equation becomes exact after multiplying it by R.
(c) Hence find the general solution of the differentialequation.
(See attached file for equations).

3. Solve the boundary-value problem, if possible.
a. y''-6y'+9y =0, y(0) =1 and y(1) = 0
b. 9y''-18y'+10y = 0 , y(0) =0 and y(pie) = 1
4. If a, b and c are all positive constants and y(x) is a solution of the differential equation ay''+by'+cy = 0, show that lim x->infinity y(x) = 0

I already solved the homogeneous portion, and I need help solving the particular solution and of course combining the two to get the entire solution to the differentialequation. Not too difficult - see attachment. Please use equation editor if possible. Thank you.
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Given that:
dMS/dt = m(MN - MS) - pMS¬

A) Solve the following differentialequation by as many different methods as you can.
(See attachment for equation)
b) There is a type of differentialequation which will always be solvable by two different methods. What type of differentialequation is it and which other method can always be used to solve it?
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