Ordinary Differential Equations/First Order Linear 2

Return to Exponential GrowthEdit

Remember the population growth problem, where  ? Now that we can solve linear equations, we can also solve variations where a factor   is added in. The new equation is  , and can be solved by the linear methods taught in the last section.

ImmigrationEdit

Lets say that 1000 people move into a city, in addition to the normal population growth. This can be interpreted by making  . This gives us a linear differential equation to solve

 

 


Step 1: Find  

 

 

Letting C=1, we get  


Step 2: Multiply through

 

Step 3: Recognize that the left hand is  

 

Step 4: Integrate

 

 

Step 5: Solve for y

 

See how the answer is a constant addition to the normal solution, as expected.


HuntingEdit

Lets say the government allows 10 animals to be killed a year. This makes  . How does this effect the solution?

 

 


Step 1: Find  

 

 

Letting C=1, we get  


Step 2: Multiply through

 

Step 3: Recognize that the left hand is  

 

Step 4: Integrate

 

 

Step 5: Solve for y

 


Mixture problemsEdit

Imagine we have a tank containing a solution of water and some other substance (say salt). We have water coming into the tank with a concentration  , at a rate of  . We also have water leaving the tank at a concentration   and rate  . We therefore have a change in concentration in the tank of

 

Thinking this through,  ,  , and   are constants, but   depends on the current concentration of the tank, which is not constant. The current concentration is   where V is the volume of water in the tank. Unfortunately, the volume is changing based on how much water is in the tank. If the tank initially has   volume, the volume at time t is  . This makes the final equation

 

which is an obvious linear equation. Lets solve it.

 

Step 1: Find  

 

 

Letting C=1, we get  


Step 2: Multiply through

 

Step 3: Recognize that the left hand is  

 

Step 4: Integrate

 

 

Step 5: Solve for y

 

Ugly, isn't it. Most of the time when dealing with real world mixture problems, you'll plug in much earlier and use numbers, which makes it easier to deal with.