# Solving a Salt Transport ODE: Find Salt After 10 Minutes

• TheForumLord
In summary, the conversation is about a homework question involving two containers with different amounts of water and salt, and the transport of water between them. The goal is to write an ODE and find the amount of salt in each container after 10 minutes. The solution involves finding eigenvalues and combining two solutions to fit the initial conditions.
TheForumLord

## Homework Statement

I've attached the relevant pictures. The question is:
Let X,Y be two containers.
At t=0, container X has 100 lt. of water with 2 kg of salt in it and Y has 100 lt. of water with 6 kg of salt.
On each t>0, the system transports water as the you can see in the picture.

In each minute t, let x(t), y(t) be the quantities of salt in X,Y in kg's.
t is measured in minutes!

You should notice that on each time, there are excatly 100 lt. in each container!

Write an ODE that gives the quantity of salt on each container as a function of time, solve it and calculate how much kg's of salt will be in the container after 10 minutes from the start of the process.

## The Attempt at a Solution

I wrote the equations this way:
x'(t)= -8x(t)/100 +2y(t)/100
y'(t) = 8x(t)/100 -8y(t)/100

We get the ODE: w' =Aw ... The eignvalues of A are: -4/100 and -12/100 ...
After I solve these two equations I get two soloutions- one for x(t) and one for y(t)...The only problem is that these soloutions don't match the data of the question...
HELP is needed!

TNX!

#### Attachments

• ODE1.jpg
2.5 KB · Views: 334
What you mean is you get a solution (x1(t),y1(t)) corresponding to the eigenvalue -4/100 and a second solution (x2(t),y2(t)) corresponding to the eigenvalue -12/100, right? The general solution is a linear combination of those two solutions. You probably need to combine them to fit your initial conditions.

I've managed to solve it on my own :)

TNX a lot anyway man

## What is an ODE?

An ODE, or Ordinary Differential Equation, is a mathematical equation that contains one or more independent variables, one or more dependent variables, and their derivatives. It describes the relationship between the variables and their rates of change.

## How is salt transported in an ODE?

In an ODE for salt transport, the independent variable represents time and the dependent variable represents the amount of salt present. The derivatives in the equation represent the rate at which salt is entering or leaving the system.

## Why is it important to solve an ODE for salt transport?

Solving an ODE for salt transport allows us to predict the amount of salt present in a system at a given time, which is important for understanding and managing salt levels in various environments, such as in bodies of water or in industrial processes.

## How can we find the amount of salt after 10 minutes?

To find the amount of salt after 10 minutes, we can plug in the value of 10 for the time variable in the ODE and solve for the dependent variable, which represents the amount of salt. This will give us the predicted amount of salt after 10 minutes.

## Are there any limitations to using an ODE for salt transport?

Yes, there are limitations to using an ODE for salt transport. The equation assumes that the rate of salt transport remains constant over time, which may not always be the case. Additionally, other factors such as temperature and pressure may also affect salt transport and may not be accounted for in the ODE.

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