Can someone tell me how to show this is linearly independent or not without using Wronskian?

Question

Not sure how to do this without using the Wronskian.

Answer 1

The set of four functions are dependent.

Note that $$ e\times e^x + (-1)\times e^{x+1} +0\times e^{x^2}+0\times sin(x) =0$$ without the coefficients being all $0$

If you through away one of $e^x$ or $e^{x+1}$, then you have a set of three independent functions.