I was wondering what the difference between fundamental sets of solutions and regular solutions is when speaking about ordinary differential equations.

Thank you.

  • $\begingroup$ Could you expand a bit? The title and body of the question don't even match. $\endgroup$ – Arnaud Mortier Nov 8 at 14:05
  • $\begingroup$ @ArnaudMortier I don't understand why the procedure for finding a fundamental set of solutions is different then solving a regular IVP $\endgroup$ – Alex.G Nov 8 at 14:07

Assuming vanishing boundary conditions, the set of all solutions is a vector space (in any other case, it is an affine space and what follows essentially still holds).

A fundamental set of solutions is a basis for that vector space.

A regular solution is just one element of the vector space; if non-zero, then it may be completed into a basis, and therefore be part of a fundamental set.

Not to be mixed up with "the" general solution, which looks like a solution but is expressed in terms of parameters : if $u_1,u_2$ together make a fundamental set, then $a_1u_1+a_2u_2$ is the general solution (where $a_1$ and $a_2$ are arbitrary parameters).

  • $\begingroup$ And for example, how was it determined that the fundamental set of solutions for y''+y'-1=tan(t) is {sin(t), cos(t)}? $\endgroup$ – Alex.G Nov 8 at 17:17
  • $\begingroup$ @Alex.G This should really be a different question. But I think that before asking this new question you can use this answer as a hint to find out. $\endgroup$ – Arnaud Mortier Nov 8 at 17:31

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