Does anyone know a resource for advanced Methods for ODEs, Integrals, etc.

[Illuminerdi]

Is there a resource that is just a walkthrough of various kinds of problems one might get and the ways to solve them?

I'm not talking about the basics from the calc and difEQ series (u substitution, partial fraction decomposition, trig substitutions, trig power reduction, integration by parts; separation of variables, integration factors, exact equations, characteristic roots, laplace transforms), but rather more advanced things.

The only examples I know are residue integration, the basic nonlinear equation (not sure what to call it, but it's d^2y/dt^2=f(y), and the trick is to substitute dF(y)/dy for f(y), multiply each side by dy/dt, and then do some multivariable chain rule tricks before getting a sloppy integral with a root as the answer), using eigenvectors to solve the characteristic equation, and the multivariable and polar transformations to solve the Gaussian integral. There have to be way more that I'm unfamiliar with, even at the level any undergraduate can understand.

Is there a database of all these techniques, especially helping for a broader understanding of how to condense these techniques into easily derivable knowledge in nice packages?

 

[ZombieFeynman]

Math methods of physics by arfken and weber.

 

[aalaniz]

Dear Illuminerdi

I just posted step-by-step notes starting for ODEs and PDEs both linear and nonlinear, of the type you will see in grad physics programs. The approach is to use Lie symmetry methods. You will see that differential equations, abstract algebra, topology all go hand-in-hand towards practical methods for differential equations.

You will also see how symmetry methods unify Lagrangian, Hamiltonian and Poisson Bracket approaches in classical physics, and how symmetry methods tie over to quantum physics. You'll see how deeply symmetry methods underlie physics.

I just posted the files in my blog under aalaniz.

I got a 36 hour MS in pure math and a PhD in theoretical physics in particles and fields. I've spent ten years dotting i's and crossing t's on techniques I never felt I truly understood. I never felt like an honest PhD as long as math seemed ad hoc and full of tricks. I finally feel honest. The material in the notes should serve as the foundations for grad physics (possibly grad math), but it has been forgotten. Schools now teach each subject in isolation.

Give the notes a look. I hope they are useful.

Cheers,

Alex