A general methodical way to solve all first degree first order

[s0ft]

Is there a single, general, solution guaranteeing method that can be applied to any first degree first order differential equations? I know there are a lot of techniques or should I say categorizations for solving these types of equations, like linear, homogeneous, Bernoulli equations, exact/inexact equations etc. But sometimes I encounter problems that seem to be unsolvable with any of the mentioned methods and do not seem obvious enough to solve by inspection. Other than numerical methods, are there any such analytical paths?

 

[Ben Niehoff]

There is no general method.

 

[LCKurtz]

A D.E. guy whose name, unfortunately, I don't remember was taking questions after his colloquium presentation. He was asked what are some of the great unsolved problems in differential equations. He went to the board and wrote$$
y' = f(x,y)$$

 

[bigfooted]

I think the most general method for first order ODE's currently is the Prelle-Singer method: IF the solution of the ODE can be expressed in terms of elementary functions, the Prelle-Singer method can find the solution.

Symmetry analysis is also a very general method, but it requires that you find a symmetry of the ODE first, which can be as hard as solving the ODE itself. It is very easy to check if a symmetry is indeed a symmetry of the ODE, so a database of often encountered symmetries can be used to solve the ODE, most first order solution methods (i.e. Bernoulli transformation) use a known symmetry to solve the problem.

 

[LCKurtz]

I think the most general method for first order ODE's currently is the Prelle-Singer method: IF the solution of the ODE can be expressed in terms of elementary functions, the Prelle-Singer method can find the solution.

Apparently that method applies only if ##y'= f(x,y)## is the quotient of two polynomials.

 

[bigfooted]

Apparently that method applies only if ##y'= f(x,y)## is the quotient of two polynomials.

yes, but they can be polynomials of elementary functions, not just polynomials in y like y'=(a+by^2)/(cy^3+dy^4). The Prelle-Singer algorithm can solve 1-ODEs with Liouvillian solutions, e.g. y'=(y+1+exp(y)*x^4)/x^2y