Prove, using Wronskian, that $e^\left(3x\right)$, $e^\left(2x\right)+x$, $e^\left(x\right)+1$ are linearly independent

Question

I am currently working on a problem that asks to prove, using Wronskian, that $e^\left(3x\right)$, $e^\left(2x\right)+x$, $e^\left(x\right)+1$ are linearly independent.

I went through the general Wronskian process, creating a matrix, differentiating each column, finding the determinants, etc.

I end up with this huge equation that looks something like this:

$(9e^\left(3x\right))$$[(e^\left(2x\right)+x)(e^x)-(e^\left(x\right)+1)(2e^\left(2x\right)+1)]$ ...... $\neq$ 0

Am I expected to simplify this entire equation to prove that it $\neq$ 0?

Or am I missing something?

Answer 1

Once you compute the matrix with an indeterminate $x$, try to substitute $x=0$, and then compute the determinant. If it is $\neq 0$, then clearly the Wronskian does not vanish identically since it is $\neq 0$ for $x=0$.

Answer 2

You should find that the Wronskian is $$ W=-2\,e^{5\,x}(e^x+3) $$ Do you see why $W\neq0$?