A general methodical way to solve all first degree first order

• s0ft
In summary, there is no single, general method for solving first degree first order differential equations. While there are various techniques and categorizations for solving these types of equations, such as linear, homogeneous, and Bernoulli equations, there are still some problems that cannot be solved by any of these methods. However, there are some analytical paths, such as the Prelle-Singer method and symmetry analysis, that can be used to solve certain types of first order ODEs if the solution can be expressed in terms of elementary functions.
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?

There is no general method.

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$$y' = f(x,y)$$

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.

bigfooted said:
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.

LCKurtz said:
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

1. What is a first degree first order equation?

A first degree first order equation is an algebraic equation that contains only one variable and the highest power of that variable is 1. It can be solved using a general methodical approach.

2. What is the general methodical way to solve a first degree first order equation?

The general methodical way to solve a first degree first order equation is to isolate the variable term on one side of the equation and all the constant terms on the other side. Then, divide both sides by the coefficient of the variable to solve for the variable.

3. Can the general methodical approach be applied to all first degree first order equations?

Yes, the general methodical approach can be applied to all first degree first order equations. It is a systematic way of solving equations and will work for any equation that meets the criteria.

4. Are there any limitations to the general methodical approach for solving first degree first order equations?

While the general methodical approach is effective for most first degree first order equations, it may not work for equations with complex expressions or variables with non-numeric coefficients. In such cases, other methods like substitution or elimination may be more suitable.

5. How can the general methodical approach for solving first degree first order equations be applied in real-life situations?

The general methodical approach for solving first degree first order equations can be applied in various real-life scenarios, such as calculating interest rates, determining the cost of items on sale, or solving for an unknown value in a scientific experiment. It is a useful tool for solving many practical problems.

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