Doubt about convergence test on differential equations

In summary, the conversation discusses the use of trial and error methods to find solutions for equations, specifically with the example of x^2 = 9 and the differential equation dy/dx + (dy/dx)^2 = 0. It is noted that while trial and error can work for simple equations, it becomes more complex with differential equations as they involve a family of functions and require multiple points to be defined. It is suggested to use numerical methods to find solutions for differential equations and that there are various approaches available. It is also mentioned that the chosen example equation may be too simple for this purpose and that allowing the slope to flip between 0 and -1 can lead to interesting results. The conversation ends with a suggestion to consider the
  • #1
jonjacson
447
38
I will try to explain this with an analogy.

Let's have this equation:

x^2 =9

And let's assume I don't know algebraic methods to solve it, so I create a list using excel with different values. And I see that if I put x=4 it doesn't work, if I put x=5 it is even worse and so on. But If I put 3.9 I see it gets closer to 9, if I put 3.8, 3.7 it gets closer and closer so finally I found x=3 just by trial and error.

My question is, let's have a differential equation:

dy/dx + (dy/dx)^2 = 0

ANd let's say I want to use the previous method to find a dy/dx in a very specific point, let's say the point (0,1). The question is, Can I be sure it will converge?

With an algebraic equation even if I don't know a method to get the general solution using radicals I am able to get solutions just by substituting values and watching if they work or not. Can I do the same with differential equations?

Just by using different straight lines with different slopes and testing if they work on a very specific point, Can I be sure the values will converge to a solution?

Maybe not and I create an immense list of values with excel and I don't see the data geting closer to any slope, i don't know.
 
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  • #2
http://en.wikipedia.org/wiki/Differential_equation

In principle, yes. In practice, it takes some clever methods. And there may be easier approaches.

The problem is, your example of x^2 = 9 has a single definite answer. (Well... I suppose -3 works also.) And there is an easy to understand single measure of how good a candidate is.

For the differential equation you then pose, the answer is not a single value but a function. Actually, a family of functions, the specific one of which is selected by specifying some boundary conditions. You are really trying to find a curve that satisfies this equation. A single point is not enough. For example, a single point can't define the slope, and your equation involves the slope.

So in general it is difficult to define a simple function that will show you how far you are away from the true answer. But you can certainly define a function that shows how far away from 0 your value of dy/dx + (dy/dx)^2 is, and then do some clever iterations to make it closer to 0. It will require some cleverness to be sure you are doing it at all possible points on your curve at the same time. But there are a variety of approaches. Here are some of the most basic ones.

http://en.wikipedia.org/wiki/Numerical_methods_for_ordinary_differential_equations

I think you may have chosen an equation that is a little too simple for your purposes. Essentially what you have written is that the square of the slope is the negative of the slope. There are basically only two choices. The slope is zero or the slope is -1. If the slope is zero you have a horizontal line. Any horizontal line will do. If the slope is -1 you have a line with a -45 degree angle. Any such line will do. So you pick the particular case with boundary conditions. For example, if you required the curve to go through the points (1,0) and (2,0) it has to be one particular horizontal line. Or if you required it to go through (-1,1) and (1,-1) then it is the line at -45 degrees through the origin.

Hmmm... Now that I think of it, if you do not require the curve to have a smooth derivative you can get very interesting results by allowing the slope to flip between 0 and -1. That would be an interesting curve. One might reject such a curve if the slope had some physical interpretation.
 
  • #3
DEvens said:
http://en.wikipedia.org/wiki/Differential_equation

In principle, yes. In practice, it takes some clever methods. And there may be easier approaches.

The problem is, your example of x^2 = 9 has a single definite answer. (Well... I suppose -3 works also.) And there is an easy to understand single measure of how good a candidate is.

For the differential equation you then pose, the answer is not a single value but a function. Actually, a family of functions, the specific one of which is selected by specifying some boundary conditions. You are really trying to find a curve that satisfies this equation. A single point is not enough. For example, a single point can't define the slope, and your equation involves the slope.

So in general it is difficult to define a simple function that will show you how far you are away from the true answer. But you can certainly define a function that shows how far away from 0 your value of dy/dx + (dy/dx)^2 is, and then do some clever iterations to make it closer to 0. It will require some cleverness to be sure you are doing it at all possible points on your curve at the same time. But there are a variety of approaches. Here are some of the most basic ones.

http://en.wikipedia.org/wiki/Numerical_methods_for_ordinary_differential_equations

I think you may have chosen an equation that is a little too simple for your purposes. Essentially what you have written is that the square of the slope is the negative of the slope. There are basically only two choices. The slope is zero or the slope is -1. If the slope is zero you have a horizontal line. Any horizontal line will do. If the slope is -1 you have a line with a -45 degree angle. Any such line will do. So you pick the particular case with boundary conditions. For example, if you required the curve to go through the points (1,0) and (2,0) it has to be one particular horizontal line. Or if you required it to go through (-1,1) and (1,-1) then it is the line at -45 degrees through the origin.

Hmmm... Now that I think of it, if you do not require the curve to have a smooth derivative you can get very interesting results by allowing the slope to flip between 0 and -1. That would be an interesting curve. One might reject such a curve if the slope had some physical interpretation.

Thanks for your answer.

Very interesting your comment about not making sense a slope if you only care about one point, I forgot that basic derivative concept, sorry.

About the examples, they were just randomly choosen.

The important thing is, this is a problem a lot more complicated than evaluating an algebraic equation, even the basic substitution method could fail and I am not even able to evaluate this in a single point because the derivative is meaningless if you don't take into account an interval.
I have a doubt about your link, Do those methods work only for ordinary equations? What about generic differential equations of n order?

Just to clarify it, when I first wrote the thread I was thinking of finding the straight line which slope made the equation right at a specific point.
 

Related to Doubt about convergence test on differential equations

1. What is a convergence test for differential equations?

A convergence test for differential equations is a method used to determine whether a series of solutions to a differential equation will approach a specific value or function as the independent variable approaches a certain point.

2. Why is it important to test for convergence in differential equations?

Testing for convergence in differential equations is important because it helps to ensure that the solutions to the equation are accurate and reliable. It also helps to determine the behavior of the solutions as the independent variable changes, which can provide insight into the overall behavior of the system.

3. What are some commonly used convergence tests for differential equations?

Some commonly used convergence tests for differential equations include the Ratio Test, the Root Test, and the Integral Test. These tests utilize mathematical techniques to determine whether a series of solutions will converge to a specific value or function.

4. What factors can affect the convergence of solutions in differential equations?

The convergence of solutions in differential equations can be affected by various factors, such as the initial conditions, the form of the equation, and the range of the independent variable. Other factors, such as the presence of singularities or nonlinearity, can also impact the convergence of solutions.

5. How can I determine which convergence test to use for a specific differential equation?

The choice of convergence test for a specific differential equation will depend on the characteristics of the equation and the desired level of accuracy. It is important to analyze the equation and consider its unique features before selecting a convergence test to ensure the most appropriate method is used.

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