The problem :

Show that the functions $\displaystyle e^{ax}$,$\displaystyle e^{bx}$ and $\displaystyle e^{cx}$ are linearly independent if $\displaystyle a \neq b \neq c$

Solution :

I calculated the wronskian, and my final answer is:

$\displaystyle e^{(a+b+c)x} \left( (b-a)c^2+(a-c)b^2+(c-b)a^2 \right)$

Clearly, $\displaystyle e^{(a+b+c)x} \neq 0$

The problem here is with $\displaystyle (b-a)c^2+(a-c)b^2+(c-b)a^2$

Since $\displaystyle a \neq b \neq c$, the following numbers can't be zero:

Now, I stopped!

- b-a
- a-c
- c-b

How I can prove that the expression $\displaystyle (b-a)c^2+(a-c)b^2+(c-b)a^2$ can't be zero so that wronskian will be non zero and hence the function are linearly independent?