Why Differential Equations

INTRODUCTION

MODELING

SOLUTIONS

MODELING EXAMPLE

TORRICELLI EXPERIMENT

SEPARATION OF VARIABLES

Given that and that can be factored into a product of two functions such as , we can always separate the variables and attempt the integration. Integrate both sides with respect to gives: . On the left side of this integral equation, we use the chain rule for a single variable along with the change of variable as follows: We let . Then recall that , which is the derivative of the inner function, times the derivative of what is inside, . Making these substitutions in the above and dividing both sides by gives us the integral equation written below. This equation is ready for integration. Of course now we must consider if the integration can be done analytically? Often it can't, and we must resort to an numerical integration.



If the original differential equation is autonomous, that is or if then it is always possible to separate the variables and attempt the integration. We will consider some examples of these types.

Example 1: .

This is obviously a product of the dependent and independent variables. So we can separate the variable by inspection and write

.
Both sides of this equation can now be integrated. And we have for a solution

or



We see from these results that there are an infinite number of solutions for this equation depending on the value of "C." If , we find that . Then our solution for this specific initial value problem is



In this particular example, we must keep in mind that initial conditions such as will not work because with the inverse the sin function only takes on values in the interval [-1,1]. This is also apparent from the . It is possible to solve this problem using the Mathematica built in differential equation solver "DSolve." Note carefully the syntax. We have a list of two items enclosed in {}, the differential equation and the initial condition, both with a double equal. Next comes the dependent variable and finally, the independent variable. We see that we got the same results as found above. The solution is the same as found above. Also, note that the plot oscillates between plus and minus one.

Example 2:

This differential equation is autonomous and therefore separable. The left side may be integrated by using the method of partial fractions. We can either do this by hand or we may use a Mathematica function.

So we see with a little help from Mathematica that the solutions are:

     .

Now if we convert this logrithmetic function to the exponential form, we arrive at a solution for y.

     

Now "" is the effective integration constant. If we rearrange and solve for y, we have

     .

When , we have

     .

Solving gives . The particular solution for this specific initial value problem is as follows:

     .

Let us again use the Mathematica built in function to solve this initial value problem. In this example we will set up the equation along with the initial value in a separate instruction. We arrive at the same answer.

Next, let us plot this solution.

                In the plot we see that there is a verticle asymtote. We can set the denominator equal to zero and solve for x to find the asymtote.

Also, looking at the original problem, and the solution we must be aware that the values of y = 0 or y = -4 are not allowed for the initial conditions, because y = 0 results in an undefined log. With , we get zero in the denominator. Differential equations often have these sublities and must be scrutinized before accepting any solution. However, notice that these are both satisfactory solutions to the original differential equation. These are equilibrium solutions. Thus the general solution consists of functions of the form along with the equilibrium solutions . We did not get the two equilibrium solutions from the results obtained just from the separation of variables and integration.

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