How to Solve a Differential Equation for Water Flow in a Cylindrical Tank

In summary, the water in a cylindrical tank leaks out at a rate proportional to the depth and the acceleration due to gravity.
  • #1
vorse
13
0

Homework Statement



Water is pumped into a cylindrical tank with cross section area A at a constant rate k, and
leaks out through a hole of area a in the bottom of the tank at the rate
αa (2gh(t))^1/2
where g is the acceleration due to gravity, h(t) is the depth of water in the tank at time t,
and α is a constant with 0.5 ≤ α ≤ 1.0. It follows that
lim h(t)
t→∞



Homework Equations



A h'(t) = k-αa (2gh(t))^1/2



The Attempt at a Solution



all i got to is [ A/ k-αa (2gh(t))^1/2 ] dh = dt

I can't seem to solve the differential equation to get h(t)
 
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  • #2
Hi vorse! :smile:

(shouldn't your "a" should be the same as "A"? oh, and have a square-root: √ :wink:)

The LHS is of the form (P + Q√h)dh … so just integrate it. :wink:
 
  • #3
A and a are different; they are both constant however, so it shouldn't matter much in the integration;


"The LHS is of the form (P + Q√h)dh … so just integrate it." what do you mean by this form?


If i let u = P+Q√h
then du = 1/2Q(h)^-1/2 right? and this substitution doesn't work. I don't have anything to substitute for the du; can you clarify a little bit?
 
  • #4
vorse said:
A and a are different; they are both constant however, so it shouldn't matter much in the integration;


"The LHS is of the form (P + Q√h)dh … so just integrate it." what do you mean by this form?


If i let u = P+Q√h
then du = 1/2Q(h)^-1/2 right? and this substitution doesn't work. I don't have anything to substitute for the du; can you clarify a little bit?


(P + Q√h)dh = Pdh + Qh1/2dh
Can't you integrate these two expressions without resorting to a substitution?
 
  • #5
well, I tried separating the denominator into the form f(x) = A/cx+d +B/dx+e, etc... i think it's called partial separating integration or something, not sure, but that didn't work out. I think I'm here is because I don't know how to integrate the following equation.
 
  • #6
vorse said:
well, I tried separating the denominator into the form f(x) = A/cx+d +B/dx+e, etc... i think it's called partial separating integration or something, not sure, but that didn't work out. I think I'm here is because I don't know how to integrate the following equation.

What
denominator?

I don't see a fraction. :confused:
 
  • #7
all i got to is [ A/ k-αa (2gh(t))^1/2 ] dh = dtsee the A is divided by ( k-αa (2gh(t))^1/2)

so, in a clearer way to write it [A / ( k-αa (2gh(t))^1/2)]dh = dt

btw, what programs are out there where I can type math equations on the comp?
 

FAQ: How to Solve a Differential Equation for Water Flow in a Cylindrical Tank

1. What is a differential equation?

A differential equation is a mathematical equation that describes the relationship between a function and its derivatives. It involves the use of derivatives, which are used to represent the rate of change of a function over time.

2. Why is it important to solve differential equations?

Differential equations are used to model many real-world phenomena, such as population growth, chemical reactions, and electrical circuits. Solving these equations helps us understand and predict the behavior of these systems, making it an essential tool in scientific research.

3. How do you solve a differential equation?

There are various methods for solving differential equations, including separation of variables, integrating factors, and power series. The approach used depends on the type of differential equation and its order (the highest derivative present).

4. Can differential equations have multiple solutions?

Yes, a differential equation can have multiple solutions. This is because there are often many different functions that satisfy the same equation. However, in most cases, we are interested in finding the particular solution that fits a given set of initial conditions.

5. Are there applications of differential equations in other fields besides mathematics?

Yes, differential equations have numerous applications in other fields, such as physics, engineering, economics, and biology. They are used to model and analyze complex systems and make predictions about their behavior. For example, in physics, they are used to describe the motion of particles, while in economics, they are used to model market trends.

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