Linear dependence/independence of functions $f(x) = x^2+1$, $g(x) = 1+x^3$ and $h(x) = \ln(1 + x)$. Questions :
Prove that the functions $f(x) = x^2+1$, $g(x) = 1+x^3$ and $h(x) = \ln(1 + x)$ are linearly independent on the interval $(0, 1)$. Can you use the Wronskian to do this?
I try this:
$$c_1 x f(x)+ c_2 x g(x) + c_3 x h(x) = 0.$$
but I don't know how to solve for the values of $c_1$, $c_2$, and $c_3$.
Thanks.

The Wronskian is \begin{align}W(f,g,h)(x)&=\begin{vmatrix}f(x)& g(x) & h(x)\\f'(x) & g'(x) & h'(x) \\f''(x) & g''(x)& h''(x)\end{vmatrix}=\begin{vmatrix}1+x^2& 1+x^3 & \ln{(1+x)}\\2x & 3x^2 & \frac1{1+x} \\2 & 6x & -\frac{1}{(1+x)^2}\end{vmatrix}\\[0.3cm]&=\left\{\text{terms without} \ln(x+1)\right\}+6x^2\ln(x+1)\end{align} which means that the Wronskian does not vanish, hence the functions are linearly independent.
