Finding the Form of L: An Example

In summary, to solve for the green function of a differential equation, you need to first solve the kernel by solving Ly = 0, where L is the equation associated with the non-homogeneous equation. In this case, Lx = x^2y'' - 2xy' + 2y = 0. By considering y = x^m, you can solve for the general solution y = Cx + Dx^2. To obtain the suggested form of L, you need to make the given equation into Ly = 0, by setting y'' - alpha^2*y = h(x). The suggested form in this case would be y = x^m.
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Homework Statement



An example in my notes.

To solve for green function of a differential equation.
eg x^2(y'') - 2x(y') + 2y = x^3ln(x)

The first thing is to solve the kernel by solving Ly = 0
and it says the form of L suggests trying y = x^m

How to I obtain the suggested form of L?
 
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  • #2
By taking the suggestion?
You are told to consider the equation Ly= 0, the "associated homogeneous" equation to your non-homgeneous equation. Since you are given the equation x2y"- 2xy'+ 2y= x3ln x, it follows that Lx= x2y"- 2xy'+ 2y= 0.


If y= xm, then y'= mxm-1 and y"= m(m-1)xm-2. Putting that into the equation, Lx= 0, you get
x2(m(m-1)xm-2- 2x(mxm-1)+ 2xm=- m(m-1)xm- 2m xm+ 2xm= 0 so either x=0 or m(m-1)- 2m+ 2= m2- 3m+ 2= (m-2)(m-1)= 0. m= 1 or m= 2 which gives you the general solution y= Cx+ Dx2.
 
  • #3
Thx... I understand how to get the general solution but how to I come up with a suggested form?

Say i was given a differential equation. For example Ly = y'' - alpha^2*y = h(x). So first i need to make the equation to Ly = 0. But what is the suggested form?
 

1. What is the purpose of "Finding the Form of L: An Example"?

The purpose of "Finding the Form of L: An Example" is to provide a practical example of how to determine the mathematical form or function of a variable L in a given problem. This is a common task in scientific research and can help in understanding relationships between variables.

2. How do you approach finding the form of L?

The approach for finding the form of L involves analyzing the problem and its variables, making observations, and using mathematical techniques such as algebra, calculus, and data analysis to determine the relationship between L and other variables.

3. What are some factors that can affect the form of L?

The form of L can be affected by various factors such as the nature of the problem, the type of data available, the assumptions made, and the mathematical techniques used for analysis. Additionally, external factors such as experimental error and limitations of the study can also impact the form of L.

4. Can the form of L change over time or in different scenarios?

Yes, the form of L can change over time or in different scenarios. This is because the relationship between variables can be complex and can change depending on various factors. It is important to regularly reassess and validate the form of L in order to ensure accurate and reliable results.

5. What are some common challenges in finding the form of L?

Some common challenges in finding the form of L include dealing with incomplete or inaccurate data, incorporating multiple variables and their interactions, accounting for external factors, and determining the most appropriate mathematical techniques to use. It is important to carefully consider these challenges and address them appropriately in order to accurately determine the form of L.

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