# Prove the following statement:

• Naeem
In summary, the sum of two solutions of a linear differential equation is also a solution, with the same p(x) and the sum of q(x) and r(x) as the non-homogeneous term. To show that a function is a solution, one must demonstrate that it satisfies the differential equation by plugging it into the equation and simplifying.

#### Naeem

Q. If y = y1 (x) is a solution of dy/dx + p(x) y = r(x) and y = y2(x) is a solution of dy/dx + p(x) y = q(x), then y = y1(x) + y2(x) is a solution of

dy/dx + p(x) y = q (x) + r(x).

I know that the a Linear differential equation is of the form,

dy/dx + p(x) y = q(x)

Any thoughts on how to proceed with this one.

Would appreciate, ideas.

Please read the guidelines for posting homework help questions.

https://www.physicsforums.com/showthread.php?t=4825

You must have had a thought on something to do, even if it is something as trivial, such as rewriting a statement in terms of a definition.

Naeem said:
Q. If y = y1 (x) is a solution of dy/dx + p(x) y = r(x) and y = y2(x) is a solution of dy/dx + p(x) y = q(x), then y = y1(x) + y2(x) is a solution of

dy/dx + p(x) y = q (x) + r(x).

I know that the a Linear differential equation is of the form,

dy/dx + p(x) y = q(x)

Any thoughts on how to proceed with this one.

Would appreciate, ideas.

HOW do you show that a given function IS a solution to a differential equation? Show us what you have done or what you DO understand about this.

d(y1+y2)/dx+P(y1+y2)=(dy1/dx+py1)+(dy2/dx+py2)
=q(x)+r(x)
since q and r are solutions of those non homogeneous differential equations

## 1. What does it mean to "prove" a statement?

To prove a statement means to provide evidence or logical reasoning that supports the truth of the statement. It involves using established principles or theories to demonstrate that the statement is valid.

## 2. Can any statement be proven?

No, not all statements can be proven. Some statements may be opinions or beliefs that cannot be proven using scientific methods. Additionally, some statements may be false or untestable, making it impossible to prove their validity.

## 3. How do you begin the process of proving a statement?

The first step in proving a statement is to clearly define the statement and its components. This involves breaking down the statement into smaller, more manageable parts and determining any underlying assumptions or premises. From there, you can use logical reasoning, evidence, and experiments to support the statement.

## 4. What are some common methods used to prove a statement?

Some common methods used to prove a statement include mathematical proofs, scientific experiments, statistical analysis, and deductive reasoning. These methods involve using established principles and data to demonstrate the truth of the statement.

## 5. Can a statement be proven without any doubt?

No, it is not possible to prove a statement without any doubt. In science, there is always a possibility for new evidence or information to emerge that could challenge the validity of a statement. However, through rigorous testing and analysis, a statement can be proven with a high level of confidence.

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