A general solution is a solution to a differential equation that satisfies all possible conditions. It includes all possible solutions to the equation. A particular solution is a specific solution to the equation that satisfies a given set of initial conditions.
To find the general solution from a particular solution, you can use the method of undetermined coefficients or variation of parameters. These methods involve finding a particular solution and then adding it to the complementary solution to get the general solution.
A homogeneous general solution is one that only contains the complementary solution, while a non-homogeneous general solution includes both the complementary solution and a particular solution. A homogeneous solution is only valid for homogeneous differential equations, while a non-homogeneous solution can be used for both homogeneous and non-homogeneous equations.
No, a particular solution is only applicable to non-homogeneous differential equations. For homogeneous equations, the particular solution will be equal to zero, and therefore cannot be used to find the general solution.
Finding the general solution allows us to find all possible solutions to a differential equation, not just a specific solution. This is important because it gives us a more complete understanding of the behavior of the system being modeled by the equation. Additionally, the general solution can be used to find the particular solution for specific initial conditions.