Is this proof for a linear differential eq correct purely mathematically

In summary, the conversation is about a proof for a "linear differential eq" and whether it is correct purely mathematically. The proof includes the use of a constant of integration, which is questioned by one of the participants. It is mentioned that the constant can be factored out to make the integration easier, but there is a concern about the value of the constant and its accuracy in the integration.
  • #1
Pellefant
37
0
Is this proof for a "linear differential eq" correct purely mathematically

I wonder if this proof is correct purely mathematically
look at (3) in the link, and you will se that they have done the following ...

[tex] \int p(x)dx = \int p(x)dx +c[/tex]

So they have put out the constant of integration before they have done the integration, can they really do that?

http://www.bio.brandeis.edu/classes/biochem102/hndDiffEq.pdf
 
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  • #2
Yes, since constant + constant = constant. They probably did that to emphasize that a constant will have to be worked out to fit some initial conditions.
 
  • #3
Yea it makes it much easier because then you don't need to think about constant of integration ... In my brain i can see that as mathematically correct if c=0 because e^0=1 ...

I am fairly sure that you can't say c1+c=c2, for a value where c isn't cero, because the value from the integration (which gives c1) can just be the exactly correct value form that integration ...

Sorry if i am concerned about stuff that don't matter, but for me it is important for my understanding ...

Kindly Pellefant ...
 

What is a linear differential equation?

A linear differential equation is a mathematical equation that involves a function and its derivatives, where the function and derivatives are only raised to the first power and are not multiplied together. It can be written in the form of y' + p(x)y = q(x), where p(x) and q(x) are functions of x.

How do you know if a proof for a linear differential equation is correct?

To determine if a proof for a linear differential equation is correct, you must carefully follow the steps of the proof and check for any errors in logic or mathematical calculations. Additionally, you can verify the results by solving the equation using different methods or checking for the solution's validity in a specific range of values.

What are some common mistakes made while proving a linear differential equation?

Some common mistakes made while proving a linear differential equation include incorrect application of derivative rules, algebraic errors, and missing or incorrect initial conditions. It is also essential to ensure that the equation is in the correct form and that the steps of the proof are followed accurately.

Can a linear differential equation have more than one solution?

Yes, a linear differential equation can have more than one solution. This is because the general solution of a linear differential equation is a family of solutions, and the specific solution depends on the initial conditions given. Therefore, there can be infinite solutions to a linear differential equation if there are different initial conditions.

What are the applications of linear differential equations in science?

Linear differential equations have various applications in science, such as modeling physical systems like circuits, chemical reactions, and population growth. They are also used in engineering to analyze and design control systems and in economics to study supply and demand. Additionally, they are used in many other fields of science, such as biology, physics, and meteorology.

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