Separation of Variables

You can algebraically manipulate an equation of the form dy/dx = g(x)/h(y) into the following:

Notice how all the x's are together on one side of the equation and all the y's are together on the other. You have performed the old algebraic method of cross multiplying. Using LiveMath you can either manipulate with the Hand-mouse, perform the rules of algebra, or you can just re-input in the new form. The algebraic method is shown below:

Now in its differential form, you can integrate each side for the solution.

Notice how the derivative is constructed on the LHS of the first Prop. To solve this equation, you must construct the derivative this way. If you were to use the Partial Derivative Op from the Palette, LiveMath would not be able to give you the answer you want.

Recall from calculus that the Leibniz notation dy/dx is used to symbolically denote the derivative of y with respect to x. You can think of it as a quotient of two differentials dy and dx. This comes from the fact that the differential dy is defined as the derivative f '(x) times the differential of the independent variable x (dx).

So:

Expl #3 Separation of Variables

Input the following:

Using the same method shown above, bring the differential equation into a separated form. Remember to separate the differential operators from their variables with a space, or a multiplication ( ie; d*x => dx )

Now the equation is ready to integrate. Since the differentials are already in the problem, you apply the Integral Op in a different way this time.

Select the equation and choose Apply. Both sides will highlight, ready for an operation to be performed on them. Rather than clicking on the Indefinite Integral Op on the Palette (this will add a second differential operator), perform the command key equivalent....$. Hold the Shift key down and type a 4.

Next, select this new equation, by clicking on the equal sign, and Simplify.

You know that if Auto-Casing is on, there will be two constants generated. One way to save a step is to simplify one side of the equation with Auto-Casing turned on and the other side with it turned off. As seen in the last example, LiveMath will combine these for you, but it may take an extra step.

In either case, when you have eliminated non-essential constants, you may want to remove the subscript which remains. To do this, you can either select the number and delete it or you can just select the whole constant, subscript and all, and type a new c. A new Prop will be generated with the subscript gone.

The disadvantage of removing the subscript is that, when the new Assumption-Prop is generated, the dynamic link is gone. You will not be able to change the original problem and have it Re-manipulate the solution. It may be better to wait until the very end to do this, or just leave the subscript attached if you plan on testing different initial terms.

Now you can solve the equation for the dependent variable y.

Do this by Isolating y, or by applying the Natural Log to each side. Below the Natural Log is applied:

Once the equation is in a separated form, finding the solution is a simple matter, if the integrations are simple, that is. As you look at other methods of solving 1st Order Equations notice how, in many cases, you will try to manipulate the equation into this form. The next example shows that, by multiplying each term by, what is called an integrating factor, you can do just that.

Expl #4 Separation of Variables #2

Input the following:

You should be able to see that this equation is not in a form that you can solve directly by integration. This is because the variables are not separated.

To place it in the correct form, you need to multiply each side by an Integrating Factor. The factor for this equation is, 1/xy:

Now the equation is in a separated form and you can integrate.

Below, follow the steps done in the last example except, apply the exponential function to both sides. This will isolate y:

The solution above is in an explicit form, but you can simplify it into a more conventional form by using the laws of exponents. Below, the constant has been relieved of its subscript to better display the problem.

From your past mathematical experience, you should know how to simplify this expression. To have LiveMath do it for you, you will learn a new technique.

If you try to simplify this equation by selecting and choosing Simplify, the expression will not change. The trick is to first select the RHS and Collect with Auto-Simplify turned OFF. This will evoke the first law of exponents where:

Applying this law to the example gives:

Since e is a constant, a constant taken to a constant power is just going to be another constant. It follows that you can then change this to a newly labeled constant containing both.

In your notebook select e and its power (c), and type a new letter. You can use any letter as long as you have defined it as a constant. Some books will use "A" and others will use "C". Below it is just re-named "c". The last manipulation is to commute the constant to the other side of the term, which is the conventional location:

Both of the last two examples illustrate the fact that whenever you separate the variables in a separable differential equation, you are using a type of Integrating Factor. In general an Integrating Factor is a function which, when multiplied on to each side of a differential equation, yields an equation where each side is recognized as a derivative.

The use of integrating factors is used repeatedly in the study of differential equations, to transform equations from a form you cannot easily solve, into a form you can. They are useful in solving some types of Homogeneous equations (the next subject) and particularly useful in solving 1st Order Linear equations.