I was wondering if you guys could help me out with a couple of differential equations problems that have been driving me crazy.
In the first, I'm told that there is a hemispherical tank of radius four feet filled with water, and that at time zero, a circular hole of diameter one inch is opened at the bottom of the tank. I'm asked to find the time it takes the tank to drain.
I have no idea how to even model this one. I can't figure out the physics involved. I'm sure it must be something simple that I'm missing, but I don't know what. I've only taken the introductory physics series, so the only equations that I know which relate to fluid flow are Bernoulli's equation and the equation of continuity. I thought maybe I could use the fact that the volume rate of flow is equal to the area of the opening times the velocity of the fluid passing through the opening, but that still leaves the problem of finding the velocity of the fluid passing through the opening. Obviously, Torricelli's Theorem doesn't apply since the velocity of the water at the top of the tank will approach the velocity of the water passing through the opening as the water level falls. And even if it did apply, I would only be able to solve for the velocity in terms of the water level, which is itself an unknown function of time.
The second problem is finding the general solution to a second order linear ODE with nonconstant coefficients. The equation is . This doesn't fit any general forms for solvable second order linear ODEs that I can find. Hopefully, there's a method that can be applied to this equation that I've missed, and one of you knows it.
Thanks in advance for your help.