Q8E - Fundamentals Of Differential Equations And Boundary Value Problems

Question

Determine whether the given functions are linearly dependent or linearly independent on the specified interval. Justify your decisions.

$\left\{x^2, x^2-1, 5\right\}$ on $\left(-\infty , \infty \right)$

Step-by-Step Solution

Answer

Thus, $\left\{{\mathbf{x}}^{\mathbf{2}}\mathbf{,} {\mathbf{x}}^{\mathbf{2}}\mathbf{-}\mathbf{1}\mathbf{,} \mathbf{5}\right\}$ are linearly dependent on $\left(\mathbf{-}\infty \mathbf{,} \infty \right)$.

Step 1:Using the concept of Wronskian

The given function is $\left\{{\mathbf{x}}^{\mathbf{2}}\mathbf{,} {\mathbf{x}}^{\mathbf{2}}\mathbf{-}\mathbf{1}\mathbf{,} \mathbf{5}\right\}.$

 

Apply the concept of Wronskian,

 $W\left[f_1,f_2,\dots ,f_n\right]=\left|\begin{array}{cccc}{f}_1\left(x\right)& f_2\left(x\right)& \dots & f_n\left(x\right)\\ f_1\text{'}\left(x\right)& f_2\text{'}\left(x\right)& \dots & f_n\text{'}\left(x\right)\\ ⋮& ⋮& & ⋮\\ {f_1}^{n-1}\left(x\right)& {f_2}^{n-1}\left(x\right)& \cdots & {f_n}^{n-1}\left(x\right)\end{array}\right|$

Therefore,

$W\left[x^2,x^2-1,5\right]=\left|\begin{array}{ccc}{x}^2& x^2-1& 5\\ 2x& 2x& 0\\ 2& 2& 0\end{array}\right|$

Solve the above equation, 

 $$\begin{gathered} \mathbf{W}\left[{\mathbf{x}}^{\mathbf{2}}\mathbf{,} {\mathbf{x}}^{\mathbf{2}}\mathbf{-}\mathbf{1}\mathbf{,} \mathbf{5}\right]\mathbf{=}{\mathbf{x}}^{\mathbf{2}}\left(\mathbf{0}\right)\mathbf{-}\left({\mathbf{x}}^{\mathbf{2}}\mathbf{-}\mathbf{1}\right)\left(\mathbf{0}\right)\mathbf{+}\mathbf{5}\left(\mathbf{4}\mathbf{x}\mathbf{-}\mathbf{4}\mathbf{x}\right) \\ \mathbf{=}\mathbf{0} \end{gathered}$$

Step 2: Check the linearly independent or dependent

The above function is equal to zero $\left(\forall \mathbf{x}\right).$

 

Therefore, $\left\{{\mathbf{x}}^{\mathbf{2}}\mathbf{,} {\mathbf{x}}^{\mathbf{2}}\mathbf{-}\mathbf{1}\mathbf{,} \mathbf{5}\right\}$ are linearly dependent on $\left(\mathbf{-}\infty \mathbf{,} \infty \right)$.