Ordinary Differential Equations/First Order Linear 2

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Return to Exponential Growth[edit]

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.

Immigration[edit]

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.


Hunting[edit]

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 problems[edit]

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.