# Best Guess for Partial Solution for Diff EQ?

• kq6up
In summary, the conversation discusses the process of finding the specific solution for a given differential equation using the method of undetermined coefficients. The conversation mentions the use of an annihilator to eliminate guesswork and the correct "guess" to use for undetermined coefficients. It also discusses the possibility of trial and error in finding the correct solution and mentions a possible alternative method, the annihilator method.
kq6up

## Homework Statement

Find the specific solution for: y''-2y'+y=xe^x+4, y(0)=1, y'(0)=1.

N/A

## The Attempt at a Solution

Since xe^x is already in the general solution of the homogeneous version of this diff eq (complementary solution), my first guess for a partial solution term would be x^2e^x (xe^x is already taken by the complementary sol). However, it no worky. The correct guess is x^3e^x. I am not sure why this term has a higher power of x then the order of the DiffEq. I would have never have guessed this. Is there a reason I should have guessed this, or in general do I have to keep increasing powers until I find one that works? I am trying to find ways to save time on an exam.

Thanks,
Chris [/B]

kq6up said:

## Homework Statement

Find the specific solution for: y''-2y'+y=xe^x+4, y(0)=1, y'(0)=1.

N/A

## The Attempt at a Solution

Since xe^x is already in the general solution of the homogeneous version of this diff eq (complementary solution), my first guess for a partial solution term would be x^2e^x (xe^x is already taken by the complementary sol). However, it no worky. The correct guess is x^3e^x. I am not sure why this term has a higher power of x then the order of the DiffEq. I would have never have guessed this. Is there a reason I should have guessed this, or in general do I have to keep increasing powers until I find one that works? I am trying to find ways to save time on an exam.

Thanks,
Chris [/B]
The best way is to eliminate the guesswork with the annihilator method. It is really pretty simple. See
http://www.utdallas.edu/dept/abp/PDF_Files/DE_Folder/Annihilator_Method.pdf

Last edited by a moderator:
LCKurtz said:
The best way is to eliminate the guesswork with the annihilator method. It is really pretty simple. See
http://www.utdallas.edu/dept/abp/PDF_Files/DE_Folder/Annihilator_Method.pdf

Ah, I vaguely remember that method from 20 years ago. Thanks for the post. According to the text my prof photocopied we are using the Method of undetermined coefficients, and I am going to assume there is guessing involved with my question above. So there is no straight forward way of knowing ahead of time with my professor's method -- just trial and error?

She assigned some painful extra credit for the whole class to repair a midterm grade, and I am thinking she wants us to just slog through it her way -- she is a bit sadistic like that :D

Thanks,
Chris

Last edited by a moderator:
Actually, the method of undetermined coefficients is the method of annihilators. That's where the "guess" comes from. On an actual problem it is pretty quick and quicker if you guess wrong for your trial solution. For a particular problem it is really quite quick. To illustrate, in your problem$$y''-2y'+y = xe^x + 4$$your annihilator on the left is ##(D-1)^2## and on the right is ##D(D-1)^2## for a combined annihilator of ##D(D-1)^4##. So you can annihilate both sides with$$\color{red}{Ae^x+Bxe^x}+Cx^2e^x+Dx^3e^x+E$$I have highlighted in red the part of that which is in the homogeneous solution. The rest of it is the correct "guess" to use for undetermined coefficients.

kq6up
Thanks,
Chris

## 1. What is a "best guess" for a partial solution for differential equations?

A "best guess" for a partial solution for differential equations is an initial approximation of the solution to a complex mathematical problem. It is often used as a starting point for further analysis and refinement.

## 2. How is a "best guess" determined for a partial solution for differential equations?

A "best guess" for a partial solution for differential equations is typically determined by using known information about the problem, such as initial values and conditions, to make an educated estimation of the solution. It can also be obtained through trial and error or by using numerical methods.

## 3. Why is a "best guess" necessary for solving differential equations?

A "best guess" is necessary for solving differential equations because these types of equations often do not have exact solutions that can be easily calculated. Therefore, an initial approximation is needed to start the process of finding a more accurate solution.

## 4. How accurate is a "best guess" for a partial solution for differential equations?

The accuracy of a "best guess" for a partial solution for differential equations depends on the complexity of the problem and the method used to obtain the approximation. In general, the more information and data available, the more accurate the "best guess" will be.

## 5. Can a "best guess" be improved upon for a partial solution for differential equations?

Yes, a "best guess" for a partial solution for differential equations can be improved upon by using more advanced mathematical techniques and algorithms. It can also be refined by incorporating new data or by repeating the process with different initial values or conditions.

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