Simple differential equation question

In summary, the conversation is discussing the possibility of converting the differential expression of dA(t,z)/dz to dI(t,z)/dz, where I(t,z) is equal to the absolute value squared of A(t,z) divided by a constant. The conversation also touches on the complexities of differentiating a complex quantity and the use of the product rule.
  • #1
n0_3sc
243
1
If I have:
[tex]\frac{dA(t,z)}{dz}[/tex]

is it possible to convert this to a differential in the form:
[tex]\frac{dI(t,z)}{dz}[/tex]
given that [tex]I(t,z)=|A(t,z)|^2/a[/tex]? (Where [tex]a[/tex] is a constant).

Any advice would help, thanks.
 
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  • #2
hi n0_3sc! :smile:

dI(t,z)/dz would be (2/a) A(t,z).dA(t,z)/dz
 
  • #3
so there's no complex conjugate anywhere?
 
  • #4
now I'm totally confused :redface:

are you saying that A is complex, but I is real? :confused:

(and in any case, I has two copies of A, and A (obviously) only has one)
 
  • #5
Sorry, A is a complex quantity so |A(t,z)|^2=A(t,z)A*(t,z)

I'm confused about differentiating that.
 
  • #6
product rule … A'A* + AA'* :wink:
 
  • #7
thanks :)
 

1. What is a simple differential equation?

A simple differential equation is a mathematical equation that involves the derivative of a function. It is used to model relationships between variables and their rates of change over time.

2. What is the difference between a simple and a complex differential equation?

A simple differential equation involves only one independent variable and its derivative, while a complex differential equation involves multiple independent variables and their derivatives. Simple differential equations are usually easier to solve compared to complex ones.

3. How do you solve a simple differential equation?

To solve a simple differential equation, you need to find the function that satisfies the equation. This can be done by integrating both sides of the equation and solving for the constant of integration. You can also use specific methods such as separation of variables or substitution to solve certain types of simple differential equations.

4. What are some real-life applications of simple differential equations?

Simple differential equations are used in many fields of science and engineering, such as physics, chemistry, biology, economics, and engineering. They are used to model and predict various phenomena, such as population growth, chemical reactions, motion of objects, and electrical circuits.

5. What are the limitations of using simple differential equations?

Simple differential equations can only model systems that have a constant rate of change, which is not always the case in real-life situations. They also assume that the system is continuous and that there are no sudden changes or disruptions. In some cases, more complex equations or models may be needed to accurately represent a system.

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