[Q]differential equation of mechanics.

[good_phy]

Hi, When i solve some lagrange equation, i encountered a lot of equation of motion that

is difficult to solve for me. They are not simeple harmonic oscillator form, even they are

partial difficult equation.

How can i solve thse equation? How do you solve?

I'm curious How can other physicist solve these difficult equation.

I do not expect exact and general method but i want useful tool to solve these problem.

[Topher925]

Well I am not a physicist but an engineer, but solve most DEs with numerical methods. Mostly the Runge-Kutta method since its easiest for me to execute. If I don't have a computer handy then I might try Laplace transforms.

[good_phy]

I'm surprise that laplace is useful tool to solve differential problem.

For solving differential equation, What kind of method does physics student use to solve?

Please give me a list including most useful method, give me a exact mathod name.

[f95toli]

As Topher925 has already pointed out: most of the DE you come across in "real life" (i.e. engineering) can't be solved analytically.
There are of course exceptions, but if are really interested in applications and already know of how to solve the "usual" equations (harmonic oscillator etc) analytically , you should probably focus on learning how to use numerical methods next.

You can spend a lot of time learning about various analytical methods but they won't be nearly as useful as a working knowledge about numerical methods.

I don't think I ever had to solve any complicated DEs analytically in the physics courses when I was a student, although we had plenty of assignments where we were expected to use numerical methods.

[statdad]

Have you looked up a resource on the separation of variables method?
Some partial differential equations in two variables have a solution u(x,y) that can be written in the form


u(x,y) = U(x) \cdot V(y)


- i.e. - it factors into two functions, each depending on one variable. Carrying out the differentiation, substituting into the partial differential equation, leads to a situation where you have


\text{Ordinary differential equation in $x$} = \text{Ordinary differential equation in $y$}


Since the two sides depend on different variables, each is equal to a constant, and you know must solve two ordinary differential equations.

I have no idea if this works for your problem, but it may be worth investigating. (You may end up with solutions defined by infinite series).