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Linear Independence and Wronskian Show that the following pair of functions are linearly independent yet their Wronskian vanishes on the given interval $$f_{1}=\begin{cases}x^{2},x\ge0\\ 0,x Sol. The given functions are $$f_{1}=\begin{cases}x^{2},x\ge0\\ 0,x 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\...

  Let          f1(x) = x,     f2(x) = x^3,     f3(x) = x^4                  f'1(x) = 1,    f'2(x) = 3x^2  f'(x) = 4x^3                  f"1(x) = 0,   f"2(x) = 6x,    f"3(x) = 12x^2 W( f 1, f 2, f 3) =                     =  Taking x common form 1st Row                =                =  Now,                                         = x(24x^4 - 18x^4)                                        = 6x^5 which is not equal to o Therefore...

Th. Prove that if the Wronskian of the functions \(f_{1}\, f_{2}\, \ldots f_{n}\) over an Interval I is non-zero, then the functions are linearly independent over I. Proof Proof: Consider the relation \[ c_{1}f_{1} + c_{2}f_{2} + \ldots + c_{n}f_{n} = 0 ...(1)\] where \( z_{1}, c_{2}, \ldots, c_{n} \) are constants. Differentiating (1) successively \( n-1 \) times with respect to \( x \), we get, \[ c_{1}f'_{1} +c_{2}f'_{2} +c_{3}f'_{3} + \ldots + c_{n} {f_{n}}^{\prime} = 0 ...(2)\] \[ c_{1}f''_{1} + c_{2}f''_{2} + \ldots + c_{n}f''_n = 0 ...(3)\] \[ c_{1}{f_{1}}^{(n-1)} + c_{2}{f_{2}}^{(n-1)} + \ldots + c_{n}{f_{n}}^{(n-1)} = 0 ...(n)\] Here The Vidoe in Youtube These \( n \) equations can be written as \[ \begin{bmatrix} f_{1} & f_{2} & \ldots & f_{n} \\ f'_{1} & f'_{2} & \ldots & f'_{n} \\ f"_{1} & f"_...