Question

Find the Wronskian of the following functions and determine whether it is linearly dependent or linearly independent on (-∞,∞).

1. {ln x, ln x²}                   ans, W=0, linearly dependent
2. {2+x, 1-x, 3+x²}           ans, W=-6, linearly independent 

Answer 1

1.

$$\{\ln x, \ln x^2\}$$

$$\{\ln x, 2\ln x\}, x>0$$

$$(\ln x)'=1/x, (\ln(x^2))'=2/x$$

$$W(\ln x, \ln x^2)=\begin{vmatrix} \ln x & \ln x^2 \\ 1/x & 2/x \end{vmatrix}$$

$$=(2/x)\ln x-(1/x)\ln x^2$$

$$=(2/x)\ln x-(2/x)\ln x=0$$

$\ln x, \ln x^2$ are linearly dependent on $(0, \infin).$

2.

$$\{2+x, 1-x, 3+x^2\}$$

$$(2+x)'=1,(1-x)'=-1, (3+x^2)'=2x$$

$$(2+x)''=0,(1-x)''=0, (3+x^2)''=2$$

$$W(2+x, 1-x,3+x^2)=\begin{vmatrix} 2+x & 1-x & 3+x^2 \\ 1 & -1 & 2x \\ 0 & 0 & 2 \end{vmatrix}$$

$$=2\begin{vmatrix} 2+x & 1-x \\ 1 & -1 \end{vmatrix}=2(-2-x-1+x)$$

$$=-6\not=0$$

$2+x, 1-x, 3+x^2$ are linearly independent on $(-\infin, \infin).$