Does $y''+ xy = 0$ admit a unique solution?

Question

Does the solution of this equation even exist? even if it does, how am I supposed to check it if it is unique? Is it necessary to find the existing solution (if at all) to check if it is unique? What if we provide an initial condition as $y(x) = a$; $y'(x) = b$; where $x$ is a particular point and $a,b$ are constants?

Answer 1

You can get infinitely many solutions of the type $\sum a_nx^{n}$. You get the relation $a_{n+3}=\frac {a_n} {(n+2)(n+3)}$ with the condition $a_2=0$. You can assign any value for $a_0$ and $a_1$ and find $(a_n)$. The series necessarily converges in a neighborhood of $0$.

Answer 2

$y''+ xy = 0$ is a homogeneous linear differential equation of order $2$. The set of all solutions if this equation is a two-dimensional real vector space.

Consequence: $y''+ xy = 0$ does not have a unique solution.