Linear differential equation; Green's function

In summary, the problem involves solving a differential equation with given boundary conditions using the method of variation of parameters. The solution should be in the form of an integral involving the known solutions y1 and y2, with a function G(x,x') that takes different forms for x' < x and x' > x. This is similar to the Green's function method described in the article provided.
  • #1
capandbells
96
0
I have this problem:

Consider the differential equation
y'' + P(x) y' + Q(x) y = 0
on the interval a &leq; x &leq; b. Suppose we know two solutions y1(x), y2(x) such that
y1(a) = 0, y1(b) ≠ 0
y2(a) ≠ 0, y2(b) = 0

Give the solution of the equation
y'' + P(x) y' + Q(x) y = f(x)
which obeys the conditions y(a) = y(b) = 0 in the form
[PLAIN]http://mathbin.net/equations/57612_0.png

where G(x,x') involves only the solutions y1, y2 and assumes different functional forms for x' < x and x' > x.

Ok, I don't really know where to begin here. My initial reaction is that I need to use variation of parameters, because it's the only method I know of that's going to give me a solution in terms of an integral, but I'm not sure if that makes sense. Anyway, if I do that, I get an expression like

[PLAIN]http://mathbin.net/equations/57613_0.png

which I don't think is right because it doesn't obey the y(a) = y(b) = 0 condition. I also don't get that "assumes different functional forms..." part. I really have no idea what I'm doing here, I'm mostly flailing aimlessly.
 
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  • #2

1. What is a linear differential equation?

A linear differential equation is a mathematical equation that involves an unknown function and its derivatives. It can be written in the form of y'(x) = f(x), where y' represents the derivative of the function y with respect to x, and f(x) is a known function of x. The equation is considered linear because the function y and its derivatives appear in a linear manner.

2. What is Green's function in relation to linear differential equations?

Green's function is a mathematical tool used to solve linear differential equations. It is a solution to the homogeneous version of the equation, where the right-hand side is equal to zero. It is defined as the inverse of the differential operator in the equation, and it can be used to find the particular solution of the non-homogeneous equation by convolving it with the function on the right-hand side.

3. How is Green's function used to solve linear differential equations?

To use Green's function to solve a linear differential equation, the equation must first be written in its standard form with a zero on the right-hand side. Then, the Green's function is determined by finding the inverse of the differential operator in the equation. Finally, the particular solution can be found by convolving the Green's function with the function on the right-hand side.

4. What are the advantages of using Green's function to solve linear differential equations?

There are several advantages to using Green's function to solve linear differential equations. Firstly, it provides a general method for solving any linear differential equation, regardless of its complexity. Secondly, it allows for the use of boundary conditions to find a particular solution. Finally, it can be used to solve non-homogeneous equations, which are often more difficult to solve using other methods.

5. Are there any limitations to using Green's function to solve linear differential equations?

While Green's function is a powerful tool for solving linear differential equations, it does have some limitations. It is only applicable to linear equations, so it cannot be used for nonlinear equations. Additionally, it may be difficult to find the Green's function for more complex equations, which can limit its usefulness. Finally, it may not provide the most efficient solution for some equations, so other methods may be more suitable in those cases.

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