Method of Variation and Green Function: d^2y/dx^2 + 4y = -e^x, 0<x<L, y(0) = y(L) = 0

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mario99

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Solve the differential equation by the method of variation of parameters, then solve it again by green function. Compare your answers. Are they the same?

[math]\frac{d^2y}{dx^2} + 4y = -e^x, 0 < x < l, y(0) = y(l) = 0[/math]
The homogenous solution is
[math]y(x) = c_1\cos 2x + c_2\sin 2x[/math]
Does the method of variation of parameters mean that I have to assume the constants are functions of x?
[math]c_1 = A(x)[/math][math]c_2 = B(x)[/math]
Then I can have two equations to solve simultaneously like this:

[math]A'(x)\cos 2x + B'(x)\sin 2x = 0[/math][math]-2A'(x)\sin 2x + 2B'(x)\cos 2x = -e^x[/math]
Is this always a valid choice?
 
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mario99 said:
Solve the differential equation by the method of variation of parameters, then solve it again by green function. Compare your answers. Are they the same?

[math]\frac{d^2y}{dx^2} + 4y = -e^x, 0 < x < l, y(0) = y(l) = 0[/math]
The homogenous solution is
[math]y(x) = c_1\cos 2x + c_2\sin 2x[/math]
Does the method of variation of parameters mean that I have to assume the constants are functions of x?
[math]c_1 = A(x)[/math][math]c_2 = B(x)[/math]
Then I can have two equations to solve simultaneously like this:

[math]A'(x)\cos 2x + B'(x)\sin 2x = 0[/math][math]-2A'(x)\sin 2x + 2B'(x)\cos 2x = -e^x[/math]
Is this always a valid choice?
Click to expand...
Think about it: if they are constants, they cannot be functions of x.

Variation of parameters takes a simpler differential equation's solutions to be a model of a more complicated differential equation's solutions by replacing the constants with functions.

I don't honestly believe you know enough of this material to actually be able to solve this problem using variation of parameters, much less using Green's functions, without simply copying the solution from another source. The last time I saw you, you were struggling with basic Calculus. I have no reason to believe that you are capable of Differential Equations. Please master the requisite material before you try to advance; you have been advised to do this many many times.

-Dan
 
Thank you topsquark for helping me.

You didn't answer my question! Do I have always to set the two equations like that when solving by the method of variation of parameters or every differential equation has different two equations? For the sake of simplicity, I will assume the former option.

Why you don't believe? I am a senior student now and it is normal to encounter this material. The Airy equation should give you a sign, however, it is a lot easier than this material. I mean that solving differential equations by Power Series is straightforward.

When I said I have to assume the constants as functions I meant the coefficients. Solving the two equations will give me:

[math]A'(x) = -\frac{B'(x)\sin 2x}{\cos 2x}[/math][math]2\frac{B'(x)\sin 2x}{\cos 2x}\sin 2x + 2B'(x)\cos 2x = -e^x[/math][math]B'(x) = \frac{-e^x}{2\frac{\sin 2x}{\cos 2x}\sin 2x + 2\cos 2x}[/math][math]A'(x) = \frac{(\frac{e^x}{2\frac{\sin 2x}{\cos 2x}\sin 2x + 2\cos 2x})\sin 2x}{\cos 2x}[/math]
What should I do next?
 
mario99 said:
You didn't answer my question!
Click to expand...
Nope. And the fact that you even asking it shows to me that you are not ready for this material. The application to a problem this simple literally follows directly, step by step, from the textbook derivation of the final formula and solving the system of equations in this case is also to be expected of a student at this level. And you can find the answer to your questions in just about any source on this, written out with all steps included.

If someone else wishes to help you, then fine. But I will not encourage you to continue to work with material you are not ready for. You've been doing it for years despite being cautioned to slow down and I haven't noted any improvement in your technique.

-Dan
 
I understand your logic, the way you are looking at me. You are saying that if I am a senior student, I will not be struggling to solve this simple differential equation. Otherwise, I am not ready for this material or course and instead I should study first the requisite. Do you think also that I am playing around? Say it please.

You may be right about the differential equation. It is simple. It is not that I don't know how to solve it! The problem is how to solve it using the method of variation of parameters. I have never used this method before. I can solve this non-homogenous differential equation in a blink of an eye if I am allowed to use the normal methods I learned in my junior years.

I have posted for you some of the steps that the book is using to solve the equation. I don't understand them and I am asking you specific questions about them but you are ignoring. Do you think that I will be able to come up with the idea of the two equations by myself? Of course no, it is from the book.

Isn't this forum about showing where you are stuck at and you will get help at?

I advise you one thing. You should forget my history. I am not the same Mario three years ago. I am not a kid any more if that what you see at me. Also forget about what I said in the post of Airy equation and open a new page.

If you are not willing to help, you should not have wasted your time and my time to reply to this post.
 
mario99 said:
I understand your logic, the way you are looking at me. You are saying that if I am a senior student, I will not be struggling to solve this simple differential equation. Otherwise, I am not ready for this material or course and instead I should study first the requisite. Do you think also that I am playing around? Say it please.

You may be right about the differential equation. It is simple. It is not that I don't know how to solve it! The problem is how to solve it using the method of variation of parameters. I have never used this method before. I can solve this non-homogenous differential equation in a blink of an eye if I am allowed to use the normal methods I learned in my junior years.

I have posted for you some of the steps that the book is using to solve the equation. I don't understand them and I am asking you specific questions about them but you are ignoring. Do you think that I will be able to come up with the idea of the two equations by myself? Of course no, it is from the book.

Isn't this forum about showing where you are stuck at and you will get help at?

I advise you one thing. You should forget my history. I am not the same Mario three years ago. I am not a kid any more if that what you see at me. Also forget about what I said in the post of Airy equation and open a new page.

If you are not willing to help, you should not have wasted your time and my time to reply to this post.
Click to expand...
I am not wasting your or my time. I am telling you the same thing that I have told you already, a countless number of times, on several different forums. Slow down and get a better grasp of the material before you move ahead. I am not saying this to impede your progress, rather, I am saying it as a professional educator. That's the job I volunteer here for and I take it seriously.

If someone else wants to help you with the problem, they are more than welcome to. But don't say that I am not trying to help you: I am, you just don't want to hear what I am saying.

And, no, I am most certainly not going to forget your history. You have implied (the same number of countless times) that all of the forums that you have been kicked off of should let you back on and give you another chance. How many chances should you get, mario99? You are still pushing to learn material that is out of your ken: you are learning, but not getting anywhere near a full picture and often have to be reminded of material that you should have mastered already. You are still asking me to forget about the "old you," and not for the first time. And why should I just forget about the Airy thread comment? It is indicative of your general attitude: "I am above the rules that hold for others. Do it my way." Have you changed? I see no evidence of it.

Many of the members that come here for help share similar flaws. Dealing with those are what we do. But you have all of them, and more, and you expect us to simply treat you like you are special? Yes, mario99, I think you are just playing around. Not with trying to learn Mathematics, which I think you are serious about, but with us.

-Dan
 
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mario99 said:
I understand your logic, the way you are looking at me. You are saying that if I am a senior student, I will not be struggling to solve this simple differential equation. Otherwise, I am not ready for this material or course and instead I should study first the requisite. Do you think also that I am playing around? Say it please.

You may be right about the differential equation. It is simple. It is not that I don't know how to solve it! The problem is how to solve it using the method of variation of parameters. I have never used this method before. I can solve this non-homogenous differential equation in a blink of an eye if I am allowed to use the normal methods I learned in my junior years.

I have posted for you some of the steps that the book is using to solve the equation. I don't understand them and I am asking you specific questions about them but you are ignoring. Do you think that I will be able to come up with the idea of the two equations by myself? Of course no, it is from the book.

Isn't this forum about showing where you are stuck at and you will get help at?

I advise you one thing. You should forget my history. I am not the same Mario three years ago. I am not a kid any more if that what you see at me. Also forget about what I said in the post of Airy equation and open a new page.

If you are not willing to help, you should not have wasted your time and my time to reply to this post.
Click to expand...
Please use Google and search the following key-words:

Solution of PDE using variation of parameters.​

You will find videos explaining the process very clearly - work through the example problems.
 
topsquark said:
I am not wasting your or my time. I am telling you the same thing that I have told you already, a countless number of times, on several different forums. Slow down and get a better grasp of the material before you move ahead. I am not saying this to impede your progress, rather, I am saying it as a professional educator. That's the job I volunteer here for and I take it seriously.

If someone else wants to help you with the problem, they are more than welcome to. But don't say that I am not trying to help you: I am, you just don't want to hear what I am saying.

And, no, I am most certainly not going to forget your history. You have implied (the same number of countless times) that all of the forums that you have been kicked off of should let you back on and give you another chance. How many chances should you get, mario99? You are still pushing to learn material that is out of your ken: you are learning, but not getting anywhere near a full picture and often have to be reminded of material that you should have mastered already. You are still asking me to forget about the "old you," and not for the first time. And why should I just forget about the Airy thread comment? It is indicative of your general attitude: "I am above the rules that hold for others. Do it my way." Have you changed? I see no evidence of it.

Many of the members that come here for help share similar flaws. Dealing with those are what we do. But you have all of them, and more, and you expect us to simply treat you like you are special? Yes, mario99, I think you are just playing around. Not with trying to learn Mathematics, which I think you are serious about, but with us.

-Dan
Click to expand...
Thank you very much topsquark for being honest and rigorous. At least now I can sleep at night.



khansaheb said:
Please use Google and search the following key-words:

Solution of PDE using variation of parameters.​

You will find videos explaining the process very clearly - work through the example problems.
Click to expand...
I am sure that you can construct the bridge for me. Because if you can't, then everybody can. Would you let them beat you?

Using google means cheating. Cheating is the easy way and I don't like it. Read my post of Airy equation and you will understand why I like to make tutors suffer!

Hint: They love to think the hard way.
 
mario99 said:
Yes, but it is boring to watch videos. I would rather discuss the method with someone in a forum.
Click to expand...
Then look up the Wikipedia page on it. Or look it up in a textbook. Or look it up in a pdf.

The method is basic in this case. The least you could do is try to replicate what your source shows you. Give it a try, post your results, and we'll talk. That's what we ask of everyone else. Check the forum rules.

-Dan
 
I have already asked you a simple question about the two equations, but you still didn't answer. Your answer may give me a big shortcut of such problems.
 
mario99 said:
I have already asked you a simple question about the two equations, but you still didn't answer. Your answer may give me a big shortcut of such problems.
Click to expand...
You don't learn the material by doing shortcuts.

Do the work and post it. Then we will help you.

-Dan
 
I have read the material. I got the idea.

Posting the work here will make it easy for others to understand the idea of the method of variation of parameters. I don't want them to know it from me. They have to earn it. This is why I won't post the work.

Would you like us to continue answering the post #1?
 
Mario in 2020 didn't care much if others stole his ideas. He would write everything he knew in one post. The new upgraded version of Mario in 2023 has learned from the past. I don't spoon feed others by my remarkable ideas. This is why I am better than other students. I don't want to say this and I am sure you will not believe me. I got a golden medal in mathematics and physics in 2022. Search for my name if you don't believe.
 
mario99 said:
Mario in 2020 didn't care much if others stole his ideas. He would write everything he knew in one post. The new upgraded version of Mario in 2023 has learned from the past. I don't spoon feed others by my remarkable ideas. This is why I am better than other students. I don't want to say this and I am sure you will not believe me. I got a golden medal in mathematics and physics in 2022. Search for my name if you don't believe.
Click to expand...
Then go somewhere else. This is not what this site is for.

-Dan
 
What part did you believe, topsquark? You know me and you know my math level better than anyone else.

I still need help to solve post #1.
 
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