# Multivariable DE and Momentum Problem

• jesusfreak324
In summary, the conversation discusses a momentum problem with a varying mass and air resistance taken into account. After some manipulations, the equation 6πnRv - mg(1 - p(water)/p(object)) = m (d/dt(v)) + v (d/dt(m)) is obtained. The individual is unsure how to separate the variables in the equation.
jesusfreak324
Ok, so I am doing a typical momentum problem with a varying mass (here dm/dt > 0). I am also taking into account air resistance (stoke's law essentially). After some manipulations I basically get:

6πnRv - mg(1 - p(water)/p(object)) = m (d/dt(v)) + v (d/dt(m))

I know I need to separate the variables but I am lost on how. Thank you for the assistance!

jesusfreak324 said:
Ok, so I am doing a typical momentum problem with a varying mass (here dm/dt > 0). I am also taking into account air resistance (stoke's law essentially). After some manipulations I basically get:

6πnRv - mg(1 - p(water)/p(object)) = m (d/dt(v)) + v (d/dt(m))

I know I need to separate the variables but I am lost on how. Thank you for the assistance!
With only a single equation in two dependent variables, you can't. Just as you cannot solve a single equation for two unknowns.

## 1. What is a multivariable differential equation?

A multivariable differential equation is a type of mathematical equation that involves multiple variables and their derivatives. It is used to model complex systems in fields such as physics, engineering, and economics.

## 2. What is the difference between a multivariable differential equation and a single variable differential equation?

The main difference is that a multivariable differential equation involves multiple independent variables, while a single variable differential equation only has one independent variable. This makes multivariable DEs more complex and challenging to solve.

## 3. How are multivariable differential equations used in momentum problems?

In momentum problems, multivariable differential equations are used to model the motion and interactions of multiple objects or particles. They can be used to calculate the rate of change of momentum and predict the future behavior of the system.

## 4. What are some common techniques for solving multivariable differential equations?

Some common techniques include separation of variables, substitution, and partial differentiation. Other methods such as Laplace transforms and numerical methods may also be used.

## 5. What are some real-world applications of multivariable differential equations?

Multivariable differential equations have many real-world applications, including in physics, engineering, economics, and biology. They are used to model and analyze systems such as fluid flow, heat transfer, chemical reactions, and population dynamics.

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