This is actually a first order PDE, since both partials are first partials. Since you have "second order" in the title, I'm wondering if you have written the equation correctly.PhyAmateur said:How to find L if the form is:
$$ (\frac{\partial L}{\partial x})^2 - (\frac{\partial L}{\partial y})^2 = -1$$
The author wrote, $$L = y + ax^2 + ..$$
but I didn't get how?
My approach was pretty simpleminded - it's been many years since I had a class on PDEs. My reasoning was that since the difference of the squares of the first partials was a constant (-1), then L(x, y) must be first-degree in both x and y, so that L(x, y) = ax + by + c.PhyAmateur said:Oh I have written the title wrongly! I will edit it right away. Can you please tell me how did you get L=ax+by+c?
DonePhyAmateur said:I can't edit the title :(, I would really appreciate if anyone from the mentors could edit my title!
A first-order partial differential equation (PDE) is a mathematical equation that involves derivatives of a function with respect to multiple independent variables. It is called "first-order" because the highest derivative involved is of first order.
The "Ax^2" form of a first-order PDE refers to a specific form of the equation where the coefficient of the highest derivative term is a function of only one of the independent variables, typically denoted as x. This form is useful for solving certain types of first-order PDEs.
In order to determine the value of L, you must first substitute the given values of the independent variables into the equation in Ax^2 form. This will result in a simpler equation with only one unknown variable, which can then be solved for L.
No, the Ax^2 form is only applicable to certain types of first-order PDEs, specifically those where the coefficient of the highest derivative term is a function of only one independent variable. Other forms and methods may be needed to solve other types of first-order PDEs.
The Ax^2 form is commonly used in fields such as physics and engineering to model and solve problems involving heat transfer, fluid flow, and other physical phenomena. It can also be applied in economics and finance to analyze and predict market trends and behaviors.