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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?

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    $\begingroup$ Presumably, the solution (if it exists) is not unique, as we don't have initial conditions. $\endgroup$ – Arthur Jul 2 at 8:18
  • $\begingroup$ But for a linear equation, if uniqueness exists at a point, it exists everywhere. $\endgroup$ – Ananya Singh Jul 2 at 8:23
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    $\begingroup$ You may have a look at the Airy equation and consider the change of variable $x\mapsto-x$. $\endgroup$ – Tom-Tom Jul 2 at 8:36
  • $\begingroup$ As (only) hinted in the answers, all related initial value (IVP) or Cauchy problems to that ODE have a unique solution, different initial values will give different solutions. $\endgroup$ – LutzL Jul 2 at 8:45
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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 $0$.

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$y''+ xy = 0$ 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.

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