Fundamental set of solutions to a differential equation

Question

Say I have a linear 2nd homogeneous ODE of the form $$y''(x)+p(x)y'(x)+q(x)=0$$ Now I know that the general solution to this will be of the form $$y(x)=c_{1}y_{1}(x)+c_{2}y_{2}(x)$$ where $\lbrace y_{1}(x),y_{2}(x)\rbrace$ form a fundamental set of solutions.

My question is (and apologies if it is a stupid one), is there only one set of fundamental solutions to a given differential equation (of the form above) or are there many?

Would someone be able to enlighten me on which case is true and why? Also, if the latter case is true, please give an example of two different fundamental sets of solutions that satisfy the same linear 2nd homogeneous ODE.

Answer 1

Assuming we're just considering linear ODEs:

• If it's first-order, we have an essentially unique fundamental solution, in that any nonzero solution is a scalar multiple of any other.
• If it's of higher order, we have infinitely many different fundamental solutions. For the second-order case, this is analogous to how there are infinitely many pairs of vectors $(v_1,v_2)$ whose linear span is $\mathbb{R}^2$
  • For example, consider the second order linear ODE $y''+y=0$.
  • The canonical "fundamental solutions" are $y_1(x)=\cos x, y_2(x)=\sin x$
  • However, if we take $y_1(x)=\cos(x+1), y_2(x)=\sin(x+1)$, we can show that any linear combination of these functions will give a solution (and vice versa, i.e. any solution can be written as such a linear combination)
  • One caveat to this non-uniqueness is that whilst the choice of $y_1,y_2$ can be done in multiple ways, the space of solutions they define is the same. The way we choose them is largely dependent on context.

Answer 2

There are infinitely many. This is just a basis for your solutions space; similar to the linear algebra, the basis is not unique, but the number of vector in the basis is the same.

Consider $$ y''-a^2y=0 $$ One fundamental set is $$ \{e^{at},\, e^{-at}\}, $$ another one $$ \{\sinh at,\, \cosh at\}. $$