Question

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

1. {x², x+1, x-3}                ans, W=8, linearly independent
2. {3e^2x, e^2x}                     ans, W=0, linearly dependent
3. {x², x³, x⁴}                    ans, W=2x^6, linearly independent

Answer 1

1.

$$W(y_1, y_2, y_3, x)=\begin{vmatrix} x^2 & x+1 & x-3 \\ 2x & 1 & 1 \\ 2 & 0 & 0 \end{vmatrix}$$ $$=2\begin{vmatrix} x+1 & x-3 \\ 1 & 1 \end{vmatrix}=2(x+1-x+3)=8\not=0$$

Therefore, the set $\{x^2, x+1, x-3\}$ is linearly independent on $(-\infin, \infin).$

2.

$$W(y_1, y_2, x)=\begin{vmatrix} 3e^{2x} & e^{2x} \\ 6e^{2x} & 2e^{2x} \end{vmatrix}$$ $$=6e^{2x}-6e^{2x}=0$$

Therefore, the set $\{3e^{2x}, e^{2x}\}$ is linearly dependent on $(-\infin, \infin).$

3.

$$W(y_1, y_2, y_3, x)=\begin{vmatrix} x^2 & x^3 & x^4 \\ 2x & 3x^2 & 4x^3 \\ 2 & 6x & 12x^2 \end{vmatrix}$$ $$=x^2(36x^4-24x^4)-x^3(24x^3-8x^3)$$ $$+x^4(12x^2-6x^2)=12x^6-16x^6+6x^6$$ $$=2x^6\not=0, except\ at\ \ x=0$$

Therefore, the set $\{x^2, x^3, x^4\}$ is linearly independent on $(-\infin, \infin).$