This is an interesting website. When I perform their substitutions I arrive at $$y \frac{dy}{dz} -y = \frac{2x - 1}{\ln | x |} : z:= \int -\frac{1}{x} dx$$ but from here their table provides no further help. Do you suppose this problem has never been analytically solved?LCKurtz said:Nonlinear equations can seem deceptively simple. This is a special case of an Abel DE of the second kind. This link may or may not be helpful:
http://eqworld.ipmnet.ru/en/solutions/ode/ode0125.pdf
joshmccraney said:This is an interesting website. When I perform their substitutions I arrive at $$y \frac{dy}{dz} -y = \frac{2x - 1}{\ln | x |} : z:= \int -\frac{1}{x} dx$$ but from here their table provides no further help. Do you suppose this problem has never been analytically solved?
A non-linear first order ordinary differential equation (ODE) is an equation that involves the derivative of a function, but the function itself is raised to a power or involves more complex operations such as multiplication or division. This makes the equation non-linear, as the rate of change is not directly proportional to the value of the function.
In a linear first order ODE, the function and its derivative only involve simple operations such as addition and multiplication by a constant. This makes the equation linear, as the rate of change is directly proportional to the value of the function. Non-linear first order ODEs, on the other hand, involve more complex operations and are not directly proportional.
Non-linear first order ODEs are commonly used in fields such as physics, engineering, and economics to model various systems and phenomena. Some examples include population growth, chemical reactions, and electrical circuits.
Solving a non-linear first order ODE can be more challenging than solving a linear one, as there is no general method that can be applied to all equations. However, some common techniques include separation of variables, substitution, and using an integrating factor. In some cases, numerical methods may also be used to approximate a solution.
While non-linear first order ODEs can accurately model many systems, they also have some limitations. These equations can become increasingly complex and difficult to solve as the number of variables and parameters increases. They also may not accurately represent systems that have chaotic or unpredictable behavior.