Dirac delta function under integral?

In summary, a member of a forum has a question about understanding a linear homogeneous first order differential equation and the solution provided in a research paper. The member is unsure about the Integrating factor and the final solution, and also has a question about the use of a delta function and theta function in the solution. They request guidance and clarification on these steps.
  • #1
chempinaki
2
0
Hello all,
I joined this amazing forum just today.I hope that my question will get answered soon.
So here it is.I am unable to understand a some steps in calculation. Please help me understand.

Here is a linear homogeneous first order differential equation
VRYGs.gif


whose solution a research paper writes to be:
ub3j0.gif


Now Integrating factor for solving the differential equation should have been:
EDAPv.gif


And the final solution according to the standard procedure of solving linear homogeneous first order differential equation should have been:
xx9Fv.gif


But in the paper solution is not the same.Please guide me if I am wrong with my justifications.
Also in the final step delta function selects only s=t prime value.I am unable understand why theta function is coming.please elaborate on this step too.Any help will be deeply appreciated. I don't come from a maths background.
 
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  • #2
somebody please address the issue.thanks in advance.i am utterly confused.
 

1. What is the Dirac delta function under integral?

The Dirac delta function under integral, also known as the Dirac measure, is a mathematical function that is defined as zero everywhere except at the origin, where it is defined as infinity. It is commonly used in mathematical analysis and engineering to represent a point mass or impulse at a specific location.

2. What is the purpose of using the Dirac delta function under integral?

The Dirac delta function under integral is used to simplify complex mathematical expressions and to represent point masses or impulses in physical systems. It allows for the integration of discontinuous functions and is used in many areas of science and engineering, including signal processing, quantum mechanics, and fluid mechanics.

3. How is the Dirac delta function under integral calculated?

The Dirac delta function under integral is calculated by using the sifting property, which states that the integral of the Dirac delta function over any interval that contains the origin is equal to 1. This means that the Dirac delta function can be represented as an infinite spike at the origin, with an area of 1 under the curve.

4. What are some applications of the Dirac delta function under integral?

The Dirac delta function under integral has many applications in physics and engineering. It is used to model point sources in electromagnetic fields, to solve differential equations in control theory, and to represent point masses in classical mechanics. It is also used in signal processing to analyze and filter signals.

5. Are there any limitations or drawbacks to using the Dirac delta function under integral?

There are some limitations to using the Dirac delta function under integral. It is a generalized function and does not have a physical interpretation, so it should only be used as a mathematical tool. Additionally, it is not defined at the origin, making it difficult to work with in some cases. It is also not a true function, as it does not have a well-defined value at any point other than the origin.

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