Find the Wronskian?

Question

Let $W$ be the Wronskian of the two linearly independent solution of the ODE

$$2y''+y'+t^2y=0; ~ t\in \mathbb{R}$$

Then, for all $t$, there exists a constant $C\in \mathbb{R}$ such that $W(t)$ is

1. $Ce^{-t}$
2. $Ce^{-t/2}$
3. $Ce^{2t}$
4. $Ce^{-2t}$

What I have done is try to solve the problem by using auxiliary equation and the find the solution. But I stuck in solving due to the presence of $t$, means the value of the auxiliary equation doesn't look good. Which option is correct? Help me out.

Answer 1

Hint: $$W'=(u'v-uv')'=u''v-uv''=\frac{-u'-t^2u}2v-u\frac{-v'-t^2v}2$$