Q25E - Fundamentals Of Differential Equations And Boundary Value Problems

Question

Let ${\mathbf{y}}_{\mathbf{1}}$ and ${\mathbf{y}}_{\mathbf{2}}$ be two functions defined on $\mathbf{(}\mathbf{-}\infty \mathbf{,}\infty \mathbf{)}$.

 

1. True or False: If ${\mathbf{y}}_{\mathbf{1}}$ and ${\mathbf{y}}_{\mathbf{2}}$ are linearly dependent on the interval $\left[a,b\right]$, then ${\mathbf{y}}_{\mathbf{1}}$ and ${\mathbf{y}}_{\mathbf{2}}$ are linearly dependent on the smaller interval $\mathbf{[}\mathbf{c}\mathbf{,}\mathbf{d}\mathbf{]}\subset \mathbf{[}\mathbf{a}\mathbf{,}\mathbf{b}\mathbf{]}$.
2. True or False: If ${\mathbf{y}}_{\mathbf{1}}$ and ${\mathbf{y}}_{\mathbf{2}}$ are linearly dependent on the interval $\left[\mathbf{a}\mathbf{,}\mathbf{b}\right]$, then ${\mathbf{y}}_{\mathbf{1}}$ and ${\mathbf{y}}_{\mathbf{2}}$ are linearly dependent on the larger interval $\mathbf{[}\mathbf{C}\mathbf{,}\mathbf{D}\mathbf{]}\supset \mathbf{[}\mathbf{a}\mathbf{,}\mathbf{b}\mathbf{]}$.

Step-by-Step Solution

Answer

(a) The given statement is true

(b) The given statement is false

Step 1: Check whether the given statement is dependent or independent

Here given $[\mathrm{c},\mathrm{d}]\subset [\mathrm{a},\mathrm{b}]$ means $[\mathrm{c},\mathrm{d}]$ in the $[\mathrm{a},\mathrm{b}]$ . So, if  ${\mathrm{y}}_1$ and ${\mathrm{y}}_2$ are dependent on the interval $[\mathrm{a},\mathrm{b}]$ then these are linearly dependent on $[\mathrm{c},\mathrm{d}]$ .

 

Therefore, the given statement is true.

Step 2: Check whether the given statement is dependent or independent

Let's take ${\mathrm{y}}_1=\mathrm{t}$ and ${\mathrm{y}}_2=\left|\mathrm{t}\right|$ are linearly dependent on $[0,\infty )$ and not in the interval $(-\infty ,\infty )$. 

 

Therefore, the given statement is false.