Q26E - Fundamentals Of Differential Equations And Boundary Value Problems

Question

Let ${\mathbf{y}}_{\mathbf{1}}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{=}{\mathbf{t}}^{\mathbf{3}}$ and ${\mathbf{y}}_{\mathbf{2}}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{=}\left|{\mathbf{t}}^{\mathbf{3}}\right|$. Are ${\mathbf{y}}_{\mathbf{1}}$ and ${\mathbf{y}}_{\mathbf{2}}$ linearly independent on the following intervals? 

(a). $\mathbf{[}\mathbf{0}\mathbf{,}\infty \mathbf{)}$

(b). $\mathbf{(}\mathbf{-}\mathbf{\infty }\mathbf{,}\mathbf{0}\mathbf{]}$

(c). $\mathbf{(}\mathbf{-}\mathbf{\infty }\mathbf{,}\mathbf{\infty }\mathbf{)}$

(d) Compute the Wronskian $\mathbf{W}\left[{\mathbf{y}}_{\mathbf{1}}\mathbf{,}{\mathbf{y}}_{\mathbf{2}}\right]\mathbf{\left(}\mathbf{t}\mathbf{\right)}$ on the interval $\mathbf{(}\mathbf{-}\mathbf{\infty }\mathbf{,}\mathbf{\infty }\mathbf{)}$.

Step-by-Step Solution

Answer

(a). In the interval $[0,\infty )$, ${\mathrm{y}}_1$ and ${\mathrm{y}}_2$ are linearly dependent.

(b). In the interval $(-\infty ,0]$, ${\mathrm{y}}_1$ and ${\mathrm{y}}_2$ are linearly dependent.

(c). In the interval $\mathbf{(}\mathbf{-}\infty \mathbf{,}\infty \mathbf{)}$, ${\mathrm{y}}_1$ and ${\mathrm{y}}_2$ are linearly independent.

(d). The solution of the interval $(-\infty ,\infty )$ is $\mathrm{W}\left[{\mathrm{y}}_1,{\mathrm{y}}_2\right]=0$.

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

Given ${\mathrm{y}}_1\left(\mathrm{t}\right)={\mathrm{t}}^3$ and ${\mathrm{y}}_2\left(\mathrm{t}\right)=\left|{\mathrm{t}}^3\right|$

Interval is $[0,\infty )$ in this interval $\left|{\mathrm{t}}^3\right|={\mathrm{t}}^3$ means ${\mathrm{y}}_1={\mathrm{y}}_2$. So ${\mathrm{y}}_1$ and ${\mathrm{y}}_2$ are linearly dependent.

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

Interval is $(-\infty ,0]$ in this interval $\left|{\mathrm{t}}^3\right|=-{\mathrm{t}}^3$ means ${\mathrm{y}}_1=-{\mathrm{y}}_2$. So ${\mathrm{y}}_1$ and ${\mathrm{y}}_2$ are linearly dependent

Step 3: Check whether the given statement is dependent or independent

Interval is $(-\infty ,\infty )$

${\mathrm{c}}_1{\mathrm{t}}^3+{\mathrm{c}}_2\left|{\mathrm{t}}^3\right|=0$

 

The above equation is true only when ${\mathrm{c}}_1={\mathrm{c}}_2=0$. Therefore ${\mathrm{y}}_1$ and ${\mathrm{y}}_2$ are linearly independent.

Step 4: Compute the Wronskian

If,

$\begin{array}{rcl}{\mathrm{y}}_1& =& {\mathrm{t}}^3\Rightarrow {\mathrm{y}}_1^{}\text{'}\\ & =& 3{\mathrm{t}}^2\\ {\mathrm{y}}_2& =& \left|{\mathrm{t}}^3\right|\Rightarrow \frac{\left|{\mathrm{t}}^3\right|}{{\mathrm{t}}^3}\times 3{\mathrm{t}}^2\\ & =& 3\frac{\left|{\mathrm{t}}^3\right|}{\mathrm{t}}\end{array}$

Then,