When solving a first-order ODE (perhaps there is also a way to extend this to a higher order ODE) for [imath]y(x)[/imath], it is possible to shift perspective and consider [imath]x[/imath] to be a function of [imath]y[/imath] by manipulating the differential forms. For example, [imath]\frac{dy}{dx} = h(y)[/imath] can be solved as [imath]\frac{dx}{dy} = \frac{1}{h(y)}[/imath], which may be easier. The moral of the story, very crudely stated, seems to be that derivatives are slopes, and we can inquire about the slope of any variable with respect to any other, regardless of which variables are "actually independent" at the end of the day.
When solving a first-order PDE such as [imath]u_t + 2xu_x = 0[/imath], one of the steps involves considering either [imath]\frac{dt}{dx}[/imath] or [imath]\frac{dx}{dt}[/imath], even though both [imath]x[/imath] and [imath]t[/imath] are independent variables, which I'm fine with due to the above paragraph. However, either of these derivatives are situated within a horizontal slice of [imath]\mathbb{R}^3[/imath], where [imath]u[/imath] is the dimension orthogonal to that slice. Thus these derivatives should technically be partials, and the "ODEs" technically single variable PDEs, and more importantly, the constant of integration encountered when we solve either of these "ODEs" should be a function of [imath]u[/imath]. In this particular problem (solved using the choices made in the above timestamped video), we would get [imath]x = K(u)e^{2t}[/imath], which produces the correct solution [imath]u = f(xe^{-2t})[/imath] after dividing through by [imath]e^{2t}[/imath] and taking [imath]K^{-1}[/imath] of both sides. In other solution methods for first-order PDEs, constants are explicitly written as functions.
1) Does thinking about the constant of integration as a function in the context of first-order PDEs always work as it does here, or is this a coincidence? If not, then why is it correct to use a genuine constant here instead of a function of the dimension not involved in the derivative? Again, I'm drawing the conclusion from the first paragraph that [imath]u[/imath] not "actually" being an independent variable makes little difference in this context.
2) For separation of variables problems, it is similarly common to shorthand partial derivatives as ordinary derivatives once things are separated. However, it again feels like the constants of integration (which are often thought of as constants of linear combinations, but can be shown to be constants of integration by solving the separated PDEs using, i.e., reduction of order) should be functions of the other variable which was separated out. Why aren't they?
Is there a systematic way to decide when integration constants in a three-dimensional problem should be functions, or does it come down to memorizing the decisions presented along with each solution method?
When solving a first-order PDE such as [imath]u_t + 2xu_x = 0[/imath], one of the steps involves considering either [imath]\frac{dt}{dx}[/imath] or [imath]\frac{dx}{dt}[/imath], even though both [imath]x[/imath] and [imath]t[/imath] are independent variables, which I'm fine with due to the above paragraph. However, either of these derivatives are situated within a horizontal slice of [imath]\mathbb{R}^3[/imath], where [imath]u[/imath] is the dimension orthogonal to that slice. Thus these derivatives should technically be partials, and the "ODEs" technically single variable PDEs, and more importantly, the constant of integration encountered when we solve either of these "ODEs" should be a function of [imath]u[/imath]. In this particular problem (solved using the choices made in the above timestamped video), we would get [imath]x = K(u)e^{2t}[/imath], which produces the correct solution [imath]u = f(xe^{-2t})[/imath] after dividing through by [imath]e^{2t}[/imath] and taking [imath]K^{-1}[/imath] of both sides. In other solution methods for first-order PDEs, constants are explicitly written as functions.
1) Does thinking about the constant of integration as a function in the context of first-order PDEs always work as it does here, or is this a coincidence? If not, then why is it correct to use a genuine constant here instead of a function of the dimension not involved in the derivative? Again, I'm drawing the conclusion from the first paragraph that [imath]u[/imath] not "actually" being an independent variable makes little difference in this context.
2) For separation of variables problems, it is similarly common to shorthand partial derivatives as ordinary derivatives once things are separated. However, it again feels like the constants of integration (which are often thought of as constants of linear combinations, but can be shown to be constants of integration by solving the separated PDEs using, i.e., reduction of order) should be functions of the other variable which was separated out. Why aren't they?
Is there a systematic way to decide when integration constants in a three-dimensional problem should be functions, or does it come down to memorizing the decisions presented along with each solution method?