First Order PDE Solution Method Issues

[sponsoredwalk]

I'd really appreciate help with two little questions relating to first order partial differential equations.

Just to quickly let you know what I'm asking, the first is about solution methods t first order PDE's & pretty much requires you to have familiarity, by name, with Lagrange's method, Charpit's Method, Jacobi's Method & Cauchy's Method of characteristics & understand the distinctions between them (discussed http://www.iitk.ac.in/math/faculty/malayb/pdenotes.pdf, http://bobbyness.net/NerdyStuff/Dif...nlinear_files/The Lagrange Charpit Method.pdf & here). Although many books unfortunately just take only one of these approaches, such as https://www.amazon.com/dp/1571460365/?tag=pfamazon01-20, here, https://www.amazon.com/dp/0821849743/?tag=pfamazon01-20 & here, other books, such as 5, 6, 7 & 8, at least let you know there are these different methods though I'm a little confused. My second question is basically about the "characteristic equations" $$\frac{dx}{P(x,y,z)} = \frac{dy}{Q(x,y,z)} = \ ... \$$, more specifically about the distinction between $$\frac{dx}{P(x,y,z)} = \frac{dy}{Q(x,y,z)}$$ & $$\frac{\frac{dx}{dt}}{P(x,y,z)} = \frac{\frac{dy}{dt}}{Q(x,y,z)} \$$, something obviously motivated by, related to, and may even answer, the first question though it seems to have taken on a life of it's own.

The first question is about solution methods for first order PDE's, as I understand it there are basically two methods: Lagrange's method, with Charpit's extension to the nonlinear case, & Cauchy's method of characteristics which is supposed to hold in both the quasilinear & fully nonlinear cases. I've only superficially studied both of these but can't really advance any further because both methods look practically the same & I'd like to know the fundamental difference between them. In 8 they develop Cauchy's Method of characteristics for a nonlinear equation by, at one stage, borrowing from results they'd established using Lagrange's method (i.e. developing something in terms of parameters then implicitly assuming parameter-independent theory), while in 5 they discuss Lagrange's method after developing Cauchy's method & do it entirely in terms of parametrizations (something you wouldn't even know was possible had you only studied from here which does everything without mentioning parametrizations). As far as I can tell the only difference is that Cauchy's method applies to both quasilinear & fully nonlinear cases while Lagrange's method applies only to quasilinear equations, though Lagrange & Charpit's method applies to the nonlinear case when you have a function of two independent variables (though 6 & 7 say that Jacobi's method is just a further extension of Charpit's method to functions of n variables though confusingly 8 says Jacobi's method is an extension of Cauchy's method, hence you see why I'm confused). I originally thought the main distinction between these methods was that Cauchy's method required parametrizations whereas the other methods ignore the parameters (i.e. whether you end up solving $$\frac{dx}{P(x,y,z)} = \frac{dy}{Q(x,y,z)}$$ or $$\frac{\frac{dx}{dt}}{P(x,y,z)} = \frac{\frac{dy}{dt}}{Q(x,y,z)} \$$), but if Cauchy's method is established using non-parameter results at key steps as I've alluded to above, & that the whole method can be established in the Jacobi case with or without parameters, then obviously this can't be the fundamental distinction. I know that Cauchy's method requires the specification of initial conditions, so maybe this is the only distinction? I don't know how to make sense of the initial condition aspect of Cauchy's method if Jacobi's method can be established both with & without initial conditions as my links would have you believe. Basically I'm hoping someone could make sense of this for me.


The second question is about the distinction between $$\frac{dx}{P(x,y,z)} = \frac{dy}{Q(x,y,z)}$$ & $$\frac{\frac{dx}{dt}}{P(x,y,z)} = \frac{\frac{dy}{dt}}{Q(x,y,z)} \$$.

This book (Page 63) refers to solving $$\frac{dx}{P(x,y,z)} = \frac{dy}{Q(x,y,z)}$$ as a shorthand for solving $$\frac{\frac{dx}{dt}}{P(x,y,z)} = \frac{\frac{dy}{dt}}{Q(x,y,z)} \$$ since they are both meant to represent the definition of a curve as the intersection of two surfaces, however there's also the idea of the first notation as being completely invalid & requiring justification through the formalism of differential forms. The essay Ten Lessons I Wish I Had Learned Before I Started Teaching Differential Equations talks about the "dishonesty involved" in the first notation & how "one should bear in mind that this misleading notation is just another way of writing an autonomous system of differential equations", which leads me to wonder whether Lagrange's method, or Lagrange & Charpit's method, is not just a 'dishonest' exposition of Cauchy's method of characteristics? If Lagrange's method is just a dishonest exposition of Cauchy's method, then how does it give a general solution not requiring in initial conditions while Cauchy's method only gives a complete integral? If it isn't then what is the distinction between these methods?

Finally any links that would clear up any of this confusion would be greatly appreciated, thanks for your time!

 

[jvicens]

Thank you for your post and for reaching out for help with your questions on first order partial differential equations (PDEs). It is clear that you have put a lot of effort into your research and understanding of these different solution methods.

To answer your first question, the fundamental difference between Lagrange's method and Cauchy's method of characteristics is that Lagrange's method applies only to quasilinear equations, while Cauchy's method can be used for both quasilinear and fully nonlinear equations. This is because Lagrange's method relies on the assumption that the equation can be written in terms of two independent variables, while Cauchy's method does not have this limitation.

As for the confusion regarding Jacobi's method, it is important to note that Jacobi's method is an extension of both Lagrange's method and Charpit's method to functions of n variables. This means that it can be used for both quasilinear and fully nonlinear equations, but it requires the use of parameters in order to do so. So while Cauchy's method and Jacobi's method can both be used for fully nonlinear equations, Cauchy's method does not require the use of parameters.

Moving on to your second question, the distinction between \frac{dx}{P(x,y,z)} = \frac{dy}{Q(x,y,z)} and \frac{\frac{dx}{dt}}{P(x,y,z)} = \frac{\frac{dy}{dt}}{Q(x,y,z)} is that the first notation is a shorthand for the second notation, as the book you referenced mentions. In other words, the first notation is an autonomous system of differential equations, while the second notation is a non-autonomous system of differential equations. The dishonesty involved in the first notation is that it does not explicitly state the time variable, which can be misleading when trying to understand the dynamics of the system.

In conclusion, Lagrange's method and Cauchy's method of characteristics are two distinct solution methods for first order PDEs, with the main difference being that Lagrange's method applies only to quasilinear equations while Cauchy's method can be used for both quasilinear and fully nonlinear equations. As for the distinction between the two notations for characteristic equations, it is important to understand that the first notation is a shorthand for the second notation and does not explicitly state the time variable, which can be misleading.

I hope