Section 2.2: Separable Equations and Tricky Calculus

Section 2.2 is entitled “Solutions to Separable Equations.”

Let’s start with the definition of a separable equation. Surprise, surprise! This type of equation can be rewritten with its independent and dependent variables separated from one another.

The book uses the following equation as their example (so I’m going to use this example as well):

N is a function and lambda is a constant. This equation is use in problems involving the half-life of radioactive elements. In this example, N is the dependent variable and t (which stands for time) is the independent variable.

Now, I shall demonstrate the art of separating equations.

Now we can go one of two ways: the physics way or the tricky calculus way.

My teacher provided the following diagram to explain why we should go the physics way:

Note: not his diagram, rather my crappy Paint version of his diagram

This is what I saw:

(Note: Just kidding. But really physics is easier to understand, never mind legitimacy)

In all reality, the tricky calculus way isn’t that bad. It’s just easier to do the physics way because that’s just how physics rolls.

Now, we separate the “N” variables from the “t” variables. Note that there are no t variables on site (besides the dt) but lambda is included in the “t” variables because it’s a constant.

And we integrate!

This type of equation is also known as an exponential function, because the solution contains an exponent.

This is the basic method for solving separable equations:

1. Separate variables

2. Integrate

3. Solve for the dependent variable.

Note that separable equations will either take the form of

Adding (or subtracting) is not separable. Tricky calculus cannot help us save the day :(

Here’s a definition from the book (throwback to last section):

General solution: “A family of solutions depending on sufficiently many parameters to give all but finitely many solutions” (30).

In other words, a general solution might not give a solution to every initial value problem thrown its way (for example, when division by zero is a factor).

Here’s another definition (and another example of a separable equation):

Newton’s Law of Cooling: “The rate of change of an object’s temperature is proportional to the difference between the temperature and the ambient temperature” (31).

By the way, the example given to show Newton’s Law of Cooling involves a can of beer. Is this is a college textbook or what?

The unsolved separable equation for Newton's Law of Cooling is as follows:

A is the ambient temperature (for instance, room temperature or outside temperature). T₀ is the initial temperature of the object. The constant k can be solved for and then you’re all good to go with solving for lots of things!

Here’s a link to someone who actually took the time to solve the Newton’s Law of Cooling equation with steps and stuff: http://www.math.wpi.edu/Course_Materials/MA1022A96/lab2/node5.html

Something to note about separable equations is that sometimes solving after integration is very difficult, and sometimes not even the trickiest of calculus can simplify our solution to just one dependent variable.

For example, y = 5x + 2 is simplified.

For non-example, the following equation cannot be simplified:

Some definitions for you:

Explicit: formula as a function in terms of the independent variable (our example shows this).

Implicit: the opposite of explicit. Our non-example shows this.

Finally, I shall show you why the separation of variables step (i.e. the physics way of solving separable equations) works.

 

In all reality, validation is never a bad thing.

Nevertheless, say we start with the following equation:

Note this is very similar to the h(y)dy = g(t)dt step we do in the physics version. 

Mathematics deems this step to be meaningless (since dy and dt by themselves don’t make much sense without integrals). I understand and appreciate the validation of the physics way, and I hope you do as well.