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first of all, I have to admit I have often used this richness of knowledge that permeates through the posts of this forum to find answers to questions that I have come across in my studies. Thanks for all! Now, I have a question to post, for the first time.

I am trying to teach myself diff. equations using the MIT OpenCourseware materials and have bumped into a statement to which I have not found a rigorous answer.

One of the ways to identify the general solution to an ODE of the form [itex]\dot{x}+p(t)x=q(t)[/itex] is by the

**sum of a particular solution and the solution to a homogeneous "version"**of the differential equation where [itex]q(t)=0[/itex].

While I can see how the solution "works" by comparing it to a solution derived from the evaluation by integrating factors and the resulting integral expression, I have not been able to prove the principle (and it's bothering me). I also understand that the linearity of ODEs allows us to treat the solution to [itex]\dot{x}+p(t)x=q(t)[/itex] as equivalent to the sum of the solutions of [itex]\dot{x}+p(t)x=q(t)[/itex] and [itex]\dot{x}+p(t)x=C*0[/itex], which is generally referred to a as the superimposition principle.

I have a hunch that the two solutions are linearly independent and sufficient to span the solution space, but I cannot come up with some formal reasoning to justify it. In other words, why does such a sum always hold? Is it true for all ODEs? Can someone provide some depth to this matter?

Many thanks for reading.