whatisreality said:Homework Statement
I have two equations.
cos(θ)wφ + sin(θ)wφ = 0 (1)
And
## \frac{w_φ}{r}## + ∂wφ/∂r = 0 (2)
Find wφ, which is a function of both r and theta.
Homework Equations
The Attempt at a Solution
I end up with two equations, having integrated. wφ=## \frac{A}{sinθ}## from (1), and wφ = Ar from (2). A is a constant of integration.
Is A the same in both equations? How do I combine the two solutions into one? I don't think you add them.
I know it isn't zero... But actually the first equation isn't correct, typed wrong.6c 6f 76 65 said:Have you typed the the first equation correct? Because if not [itex]w_φ=0[/itex], and for the rest: remember that when you differentiate / integrate with respect to [itex]r[/itex] you may have lost a function of θ.
How about writing that first equation again in its corrected form, in its entirety .whatisreality said:I know it isn't zero... But actually the first equation isn't correct, typed wrong.
Should be sin(θ) multiplied by the partial differential of wφ with respect to theta
Sorry. Here it is:SammyS said:How about writing that first equation again in its corrected form, in its entirety .
A separable PDE is a type of differential equation that can be separated into individual equations for each of the variables involved. This allows for the solution to be found by solving each individual equation separately.
A non-separable PDE cannot be separated into individual equations for each variable and therefore requires more complex methods to solve. Separable PDEs are often easier to solve and have more readily available solutions.
Separable PDEs are commonly used to model physical phenomena such as heat flow, fluid dynamics, and diffusion. They are also used in financial modeling and other fields that involve continuous change over multiple variables.
To solve a separable PDE, the equation is first separated into individual equations for each variable. These equations are then solved separately using appropriate methods, such as integration or separation of variables. The solutions are then combined to find the overall solution to the PDE.
Separable PDEs are commonly used in engineering and physics to model heat and mass transfer, fluid flow, and other physical processes. They are also used in economic and financial modeling to study population growth, resource allocation, and other systems that involve continuous change over multiple variables.