Trivial solution of a differential equation I have the following ODE
$$u''(x)+4x^{3/2}\ln x\,u'(x)+8x\,u(x) = 0$$
Can I say that $u(x)$ has a trivial solution when $x = 0$?
I am a little confused as to what trivial solution means when it comes to differential equations. 
 A: A trivial solution is just only the zero solution and nothing more. In ordinary differential equations, when we way that we are looking for non-trivial solutions it just simply means any solution other than the zero solution. In your example, the trivial solution is
$$u(x) = 0,\qquad \textit{for all x in domain of interest}$$
and any solution other than this is a non-trivial solution.
A: A trivial solution is/are solutions which can just satisfy the differential equation regardless of satisfying the boundary conditions. I am not sure if this is a compact definition for trivial solutions or not but I am chiefly right. take Blasuis equation as an example which no exact analytic closed form solution has been proposed so far for such an equation.
$$f(3)+ff(2)=0$$
this equation has three boundary conditions which two of are at the zero point and the last is at the infinity. $f=k$, $f=kx$ are trivial solutions for this equation. but none of them satisfies the boundary conditions. as a matter of fact these trivial solutions do have especial meanings but currently, the nature of such solutions has not been comprehensively understood. 
I have seen that these trivial solutions can somehow show themselves into the final exact closed form solution but even many times authors have not mentioned them at all. In my perspective, these trivial solutions are somehow referred to the asymptotic behavior of the equation but this expression is yet to be proven. sometimes the general solution is a function of these trivial solutions and sometimes these trivial solutions appear themselves as an addition in the general solution (or sometimes multiplication). As a matter of fact in the case on nonlinear differential equations, the complication more rises as no one yet knows about the nature of such equations. even there are many class of linear equations which we still have an elementary knowledge about. my notion is that these complexities comes from this fact that we still have a very elementary knowledge in the science of topology. Many and many topological studies are referred to the algebraic invariants while topology should be referred to the geometrical invariants. In spite of having some hypergeometrical spaces which are already introduced to proceed with understanding the fact of topology, but yet, in my perspective, a pure algebra can hide some facts in topological concepts! by the way.
I hope that this short description can reveal some facts to you. 
Good luck       
