Constants at the end of the Frobenius method

In summary, the conversation discusses the concept of reducing two recursive relations using the Frobenius method. It explains that the constants in the solution only depend on the initial or boundary conditions, and that different initial conditions can result in different solutions. The example of ##\left(y'\right)^2 -4y = 0## is used to illustrate this, where the value of ##c## is determined by knowing the value of ##y(x_0)##.
  • #1
ABearon
5
1
TL;DR Summary
are c1 and c2 just random coefficients
I'm having a hard time grasping the concept of reducing the two recursive relations at the end of the frobenius method.

For example, 2xy''+y'+y=0
after going through all the math i get
y1(x) = C1[1-x+1/6*x^2-1/90*x^3+...]
y2(x) = C2x^1/2[1-1/3*x+1/30*x^2-1/630*x^3+...]

I know those are right, and I know we solve for what's inside the bracket by taking a C0 out. I'm just trying to clarify that, since c0 is different for each term and since it is arbitrary we can just write c1 for y1 and c2 for y2. I want to make sure this is where the c1 and c2 come from and not from trying to take out a c1 in the y1 brackets and c2 in the y2 brackets.
 
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  • #2
The constants only depend on the initial or boundary conditions. A certain solution is a certain flow through a vector field: different initial conditions make different flows although the vector field does not change.

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Example: ##\left(y'\right)^2 -4y = 0## has the solution ##y=x^2##. But this is only half the truth. All functions ##y(x)=x^2+c## are solutions, too. And we find the value of ##c## by knowing some actual value of ##y(x_0)##. More or higher derivatives result in more initial conditions. The point ##x_0## is often chosen to be ##x_0=0## but could by any other. Say in my example we have ##y(-3)=16##, then ##y(-3)=(-3)^2+c=16## which get's us ##c=7##.
 

Related to Constants at the end of the Frobenius method

1. What is the Frobenius method?

The Frobenius method is a mathematical technique used to solve differential equations with singular points, or points where the coefficients of the equation become infinite. It is named after the mathematician Ferdinand Georg Frobenius.

2. How does the Frobenius method work?

The Frobenius method involves assuming a solution to the differential equation in the form of a power series, and then solving for the coefficients of the series. This allows for solutions to be found even when the coefficients are not constant.

3. What are constants at the end of the Frobenius method?

Constants at the end of the Frobenius method refer to the coefficients of the power series that are not determined by the differential equation. These constants are typically denoted as C1, C2, etc. and their values can be determined by applying boundary conditions to the solution.

4. Why are constants at the end of the Frobenius method important?

Constants at the end of the Frobenius method are important because they allow for a general solution to be found for the differential equation. Without these constants, the solution would only be valid for a specific set of boundary conditions.

5. What are some applications of the Frobenius method?

The Frobenius method has many applications in physics, engineering, and other fields where differential equations are used to model physical systems. It is commonly used to solve equations in quantum mechanics, fluid mechanics, and heat transfer, among others.

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