information:extraction, quality:correction, linguistic:voice_change
The guessing method

When all other methods for solving an ODE fail, or in the cases where we have some intuition about what the solution to a DE might look like, it is sometimes possible to solve a DE simply by guessing the solution and validating it is correct. To use this method, we simply guess a solution to the differential equation, and then plug the solution into the differential equation to validate if it satisfies the equation. If it does then we have a particular solution to the DE, otherwise we start over again and try another guess. For instance we could guess that the solution to a DE has the form: y = A e α t {\displaystyle y=Ae^{\alpha t}} since this is a very common solution that physically behaves in a sinusoidal way.

In the case of a first order ODE that is non-homogeneous we need to first find a solution to the homogeneous portion of the DE, otherwise known as the associated homogeneous equation, and then find a solution to the entire non-homogeneous equation by guessing. Finally, we add both of these solutions together to obtain the general solution to the ODE, that is:

general solution = general solution of the associated homogeneous equation + particular solution {\displaystyle {\text{general solution}}={\text{general solution of the associated homogeneous equation}}+{\text{particular solution}}}