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

Meaning
Messages
3
Reaction score
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
 
Physics news on Phys.org
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.
 
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
 
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...
 
Thread 'Direction Fields and Isoclines'
I sketched the isoclines for $$ m=-1,0,1,2 $$. Since both $$ \frac{dy}{dx} $$ and $$ D_{y} \frac{dy}{dx} $$ are continuous on the square region R defined by $$ -4\leq x \leq 4, -4 \leq y \leq 4 $$ the existence and uniqueness theorem guarantees that if we pick a point in the interior that lies on an isocline there will be a unique differentiable function (solution) passing through that point. I understand that a solution exists but I unsure how to actually sketch it. For example, consider a...
Back
Top