Is a solution of a differential equation a function of its parameters?

In summary, the electric field solution, E, is a function of the conductivity, sigma, in general, as determined by the parameters and initial and boundary conditions in the system of linear differential equations, such as the Maxwell equations.
  • #1
Meaning
3
1
Hi everyone,

Imagine I have a system of linear differential equations, e.g. the Maxwell equations.

Imagine my input variables are the conductivity $\sigma$. Is it correct from the mathematical point of view to say that the electric field solution, $E$, is a function of sigma in general, E(r,t,sigma)?

Thank you
 
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  • #2
Yes. Unless something weird is going on, the value of the solution at a particular point and time is determined by the parameters and the initial and boundary conditions.
 
  • #3
pasmith said:
Yes. Unless something weird is going on, the value of the solution at a particular point and time is determined by the parameters and the initial and boundary conditions.

Thank you
 
  • #4
Basically, any differential equation describes a family of curves, surfaces, volumes...

It's the boundary conditions and initial parameters that fix it to one curve, surface or volume...
 

1. What is a solution of a differential equation?

A solution of a differential equation is a function that satisfies the given equation when substituted into it. It represents the relationship between the independent and dependent variables in the equation.

2. Is a solution of a differential equation unique?

No, a solution of a differential equation is not always unique. There can be multiple functions that satisfy the equation and represent the relationship between the variables. However, certain conditions or constraints may be needed to determine a unique solution.

3. Can a solution of a differential equation depend on its parameters?

Yes, a solution of a differential equation can depend on its parameters. These parameters are constants that affect the behavior of the solution and can be adjusted to find different solutions for the same equation.

4. How do parameters affect the solution of a differential equation?

Parameters can affect the solution of a differential equation in various ways. They can change the shape, size, or behavior of the solution, and they can also determine the stability or instability of the solution.

5. Can a solution of a differential equation change with different initial conditions?

Yes, a solution of a differential equation can change with different initial conditions. The initial conditions provide the starting point for finding the solution, and changing them can lead to different solutions that satisfy the same equation.

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