First Order Differential Equation for Mixture Problem

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In summary, the problem involves a mixture with a rate of 10 gallons per minute going in and 4 gallons per minute going out. The tank starts at 150/300 gallons with a 1/4 lb/gal mixture. 5 gallons per minute are coming in at a 1/3 lb/gal mixture, while the other 5 gallons are clean water. The given equation is used to find the rate of change of the mixture in the tank. Integration factor is used to solve the problem.
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Outlaw747
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Homework Statement


[tex]\frac{dA}{dt}[/tex]=[tex]\frac{5}{3}[/tex] - [tex]\frac{4A}{150+2.5t}[/tex] lb/min
so basically dA/dt = rate in - rate out

Homework Equations


This is basically a mixture problem. There is 10 gals/min going in and 4gals/min going out. Tank starts at 150/300 gallons with 1/4 lb/gal mixture. 5gal/min are coming in at 1/3 lb/gal mixture, other 5 is clean water.

The Attempt at a Solution


Derived this equation, tried to manipulate it algebraically, but not really sure how to proceed. Can't seem to separate the equation and integrate, all I really know is integration factors and substitution. Any help would be great. Thanks.
 
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  • #2
Nevermind, got it using integration factor.
 

1. What is a first order differential equation?

A first order differential equation is an equation that involves the derivative of an unknown function with respect to a single independent variable. It can also be described as an equation that relates the rate of change of a function to its current value.

2. How do you solve a first order differential equation?

To solve a first order differential equation, you can use a variety of methods such as separation of variables, integrating factors, or the method of undetermined coefficients. The specific method used depends on the form of the equation and its variables.

3. What is the purpose of solving first order differential equations?

Solving first order differential equations is important in many scientific fields, including physics, engineering, and economics. They can be used to model various real-world situations and predict how a system will behave over time.

4. Can you provide an example of a real-world application of solving first order differential equations?

One example is using first order differential equations to model population growth. By setting up an equation that relates the rate of change of a population to its current size, we can predict how the population will change over time and make informed decisions about resource management.

5. Are there any limitations to solving first order differential equations?

Yes, there are certain types of differential equations that cannot be solved analytically. In these cases, numerical methods must be used to approximate a solution. Additionally, the initial conditions of the equation must be known in order to solve it, which may not always be available in real-world scenarios.

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