Inhomogeneous equations are differential equations that have a non-zero constant term on the right-hand side. This term is usually referred to as the forcing term and it causes the solution to deviate from a simple exponential function.
The most common method for solving inhomogeneous equations is the method of variation of parameters. This method involves finding a particular solution by assuming a solution in the form of a linear combination of the solutions of the corresponding homogeneous equation, and then using this particular solution to find the general solution.
The principle of variation of parameters states that if we have a linear differential equation of the form y' + p(x)y = q(x), then the general solution can be found by adding a particular solution (obtained by assuming a solution in the form of a linear combination of the solutions of the corresponding homogeneous equation) to the general solution of the homogeneous equation.
The advantage of using variation of parameters is that it allows us to find a particular solution without having to guess the form of the solution. It also provides a general method for solving inhomogeneous equations, rather than having to use specific techniques for different types of forcing terms.
No, variation of parameters is only applicable for linear differential equations. For non-linear equations, other methods such as separation of variables or Laplace transform may be used to find a solution.