Effect of wronskian on the solution of a differential equation

Question

As far as my understanding goes, the Wronskian $W(t)$ for a second order homogenous differential equation with continuous coefficients can help us govern whether the solutions will be linearly dependent or independent. Suppose we consider $t^2X''-3tX'+3X=0$ which has two solutions $X_1(t)=t^3$ and $X_2(t)=|t^3|$. Clearly, these two solutions are linearly independent over $R$. But also $W(X_1,X_2)=0$ which implies that either the solutions are identically zero or are linearly dependent. How come a contradiction is arising in this case?

Answer 1

There is no contradiction since all the theorem about the Wronskian and solutions to ODE are stated and proved for the case when $$ X''+a(t)X'+b(t)X=0 $$ and $a(t)$ and $b(t)$ are continuous. If you rewrite your equation in this form you'll get the coefficients that are not even defined at $t=0$, hence the theorems that you studied do not work for any potential solutions defined at the point $t=0$. If you restrict your attention to $t>0$ or $t<0$ then, as expected, you will find no contradiction.

Answer 2

Actually, the equation has the solutions $at^3+bt$, with $a,b\in\mathbb R$. So the Wronskian is nonzero.

You are completely correct with your first sentence.