# What exactly do they want?

• Tom McCurdy
In summary, for the given differential equation y''+25y=–1xcos(5x), the particular solution using the Method of Undetermined Coefficients is A*x*sin(5*x)+B*x*cos(5*x)+C*cos(5*x)+D*sin(5*x), where A, B, C, and D are undetermined coefficients. When using solutions of the form (Ax+B)cos(5x)+ (Cx+ D)sin(5x), for this particular equation, the solution should be multiplied by x to account for the presence of solutions to the homogeneous equation.

## Homework Statement

Set up but do not solve for the appropriate particular solution yp for the differential equation

y''+25y=–1xcos(5x)

using the Method of Undetermined Coefficients (primes indicate derivatives with respect to x).

## Homework Equations

Possible Derivatives
• sin(x)
• cos(x)
• xsin(x)
• xcos(x)

## The Attempt at a Solution

This seems like the solution should simply be

$$A*x*sin(5*x)+B*x*cos(5*x)+C*cos(5*x)+D*sin(5*x)$$

What solution are you talking about? The problem said "Set up but do not solve for the appropriate particular solution yp for the differential equation
y''+25y=–1xcos(5x)
using the Method of Undetermined Coefficients"

Normally, to get a solution of the form "x cos(5x)" you would use (Ax+B) cos(5x)+ (Cx+ D) sin(5x). However, in this case cos(5x) and sin(5x) are solutions to the homogeneous equation. Multiply what you would normally use by x: (Ax2+ Bx)cos(5x)+ (Cx2+ Dx)sin(5x).

## 1. What is the purpose of their research?

The purpose of their research can vary depending on the specific study or experiment being conducted. Generally, scientists aim to expand knowledge and understanding in a particular field, develop new technologies or treatments, or solve a specific problem or question.

## 2. Who are "they" in this context?

"They" can refer to a group of researchers, scientists, or experts in a particular field who are conducting the research. It could also refer to a specific organization or institution that is funding or overseeing the research.

## 3. How do they determine the methods used in their research?

The methods used in research are often determined by the specific research question or hypothesis being tested. Scientists may also consider ethical considerations, available resources, and previous research in determining the most appropriate methods for their study.

## 4. What is the significance of their findings?

The significance of a study's findings can vary depending on the research question, but generally, it contributes to our understanding of a particular phenomenon or problem. It may also have real-world applications or implications for future research in the field.

## 5. How do they ensure the validity and reliability of their results?

To ensure the validity and reliability of their results, scientists use a variety of methods such as controlling variables, replicating experiments, and peer-review. They also follow established protocols and ethical standards in their research to ensure the accuracy and credibility of their findings.

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