# Help with a relatively simple first order, first rank DE

• LennoxLewis
In summary, the conversation discusses a problem involving two cubes of water connected by a tube, with a pump that pumps a constant amount of water from one cube to the other. The goal is to derive a formula for the difference in water level as a function of time. The conversation skips the first two paragraphs and goes straight to the math part, where a mistake is found in the derivation of the differential equation. The error is in the sign of 0.1 * delta h, which affects the solution of the problem. The conversation suggests checking all signs in the equation and clearly defining delta h to ensure the solution is accurate.
LennoxLewis
Here is the problem. Skip the first two paragraphs to get to the pure math part.

There are two cubes of water, both with area of 2 x 2 dm^2. Via the bottom they are connected by a tube, and the flow of water is (surprise, surprise) proportional to the difference in water level between the two, by 0.1 * delta h. In addition to that, a constant amount of 1.0 liter / min is pumped from one to the other. At t=0, the pump is turned on when h1 = h2 = 10 dm. Derive a formula that describes delta h as function of t.

I started with: Dv1 / dt = A * dh1 / dt = 0.1*delta h - 1.0, and a similar equation for h2.
Substracting these two [i know this is risky, but i didn't know what else to do], i get:
A* d(h1-h2) / dt = A * d(delta h)/dt = 0.2*delta h - 2.0. Substracting A gives:

The math part:

d(delta h) / dt - 0.05 delta h = 0.5

Let's make things easier by renaming delta h to "x":

dx/dt - 0.05x = 0.5

I'm using a standard solution-formula, which says that for:
dx/dt + a*x = F(t), the solution is:

x(t) = exp(-a*t) * integrand (F(t')*exp(a*t')) + c*exp(-a*t)

Using this formula, i get: x(t) = -10 + c* exp(0.05*t).

Obviously, this problem would lead to a negative exponential as solution, or the difference in water level would go to infinity as time goes by!

Where do i miss a minus sign?

I skipped straight to the math part and found no problems, so I think the problem is in the physics (or, more accurately, the mathematical model of the physical situation) after all.

You should check all your signs in the equation you are writing down. In particular pay attention to the sign in front of the 0.1 * delta h. Clearly define what delta h is (is it always the height of #1 - the height of #2, so h1 - h2, for example) and convince yourself that if h1 > h2 then your differential equation ensures that delta h is positive (and if h1 < h2 then delta h is negative).

Thanks CompuChip, i just checked it and the error is indeed in the derivation of the DE, indeed the minus sign of the 0.1 * delta h.

## 1. What is a first order, first rank differential equation?

A first order, first rank differential equation is a mathematical equation that involves a function and its first derivative. It is commonly used to model physical phenomena and can be solved using various methods such as separation of variables or integrating factors.

## 2. How do I solve a first order, first rank differential equation?

There are several methods for solving a first order, first rank differential equation. The most common methods include separation of variables, integrating factors, and using the method of undetermined coefficients. It is important to choose the most appropriate method based on the specific equation and initial conditions given.

## 3. What are some real-world applications of first order, first rank differential equations?

First order, first rank differential equations have many real-world applications in fields such as physics, chemistry, biology, and engineering. They can be used to model population growth, radioactive decay, heat transfer, and many other physical processes.

## 4. Can I use a computer to solve a first order, first rank differential equation?

Yes, there are many software programs and online tools available that can solve first order, first rank differential equations. These tools use numerical methods to approximate the solution and can handle more complex equations that may not have analytical solutions.

## 5. What is the importance of understanding first order, first rank differential equations?

First order, first rank differential equations are fundamental in many areas of science and engineering. They can be used to model and predict the behavior of physical systems, making them essential for understanding and solving real-world problems. Additionally, an understanding of differential equations is necessary for advanced study in mathematics and related fields.

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