Show that the following pair of functions are linearly independent yet Wronskian vanishes on the given interval

Linear Independence and Wronskian

Show that the following pair of functions are linearly independent yet their Wronskian vanishes on the given interval

$$f_{1}=\begin{cases}x^{2},x\ge0\\ 0,x<0\end{cases}$$, $$f_{2}=\begin{cases}0,x\ge0\\ x^{2},x<0\end{cases}$$

Sol. The given functions are

$$f_{1}=\begin{cases}x^{2},x\ge0\\ 0,x<0\end{cases}$$, $$f_{2}=\begin{cases}0,x\ge0\\ x^{2},x<0\end{cases}$$.

We want to prove that $$f_{1}$$, $$f_{2}$$ are L.I. For this purpose, we shall show that

...(1)

if $$c_{1}f_{1}+c_{2}f_{2}=0$$ for all $$x\in\mathbb{R}$$, then each of $$c_{1}$$, $$c_{2}$$ is zero.

For this purpose, we use $$x=-1.1$$. Now $$f_{1}(1)=1$$, $$f_{2}(1)=0$$, and $$f_{1}(-1)=0$$, $$f_{2}(-1)=1$$. Substituting these values in (1), we get, $$c_{1}\cdot1+c_{2}\cdot0=0\Rightarrow c_{1}=0$$ and $$c_{1}\cdot0+c_{2}\cdot1=0\Rightarrow c_{2}\rightarrow0$$

We have

$$c_{1}f_{1}+c_{2}f_{2}=0\Rightarrow c_{1\frac{3}{3}}=c_{2}=0$$

Hence $$f_{1}$$, $$f_{2}$$ are linearly independent on the given interval. Now, we find the Wronskian of the given functions.

Two cases arise.

Case I. When $$x\ge0$$ we have,

$$f_{1}^{\prime}=2x$$, $$f_{2}^{\prime}=0$$

$$W(f_{1},f_{2})=|\begin{matrix}f_{1}&f_{2}\\ f_{1}^{\prime}&f_{2}^{\prime}\end{matrix}|=|\begin{matrix}x^{2}&0\\ 2x&0\end{matrix}|=0$$

Case II. When $$x<0$$ we have, $$f_{1}^{\prime}=0$$, $$f_{2}^{\prime}=2x$$

$$W(f_{1},f_{2})=|\begin{matrix}f_{1}&f_{2}\\ f_{1}^{\prime}&f_{2}^{\prime}\end{matrix}|=|\begin{matrix}0&x^{2}\\ 0&2x\end{matrix}|=0$$

Combining the results of cases I and II, we get, $$W(f_{1},f_{2})=0$$ for all $$x$$.

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