An (explicit) first-order linear ODE has the form If , linear ODE is called homogeneous, otherwise is called inhomogeneous.
Theorem (homogeneous case)
If is continuous, the general solution of with is given by The proof of this theorem is by assuming be any solution and consider . Then calculate .
Theorem (inhomogeneous case)
Suppose and are continuous.
A particular solution of is
The general solution of is
Examples
We can find out that we just need to find out one particular solution then can find out the general solution.
For the example , we can easily find out the solution , and that is the particular solution. So then, we can find out that is the general case for .
Of course the variation of parameters also works in this case. Setting , we can find Then the general solution is .
The Linear Algebra Aspect
definitions
The set of real-valued functions on a given domain is often denoted by . It forms a vector space over with respect to the 'point-wise' operations A set of functions is called a subspace if and is closed w.r.t. the vector space operations i.e. implies and implies for all .
The most important difference between and is that is infinite-dimensional.