Q31E - Fundamentals Of Differential Equations And Boundary Value Problems

Question

Determine whether the following functions can be Wronskians on $\mathbf{-}\mathbf{1}\mathbf{<}\mathbf{t}\mathbf{<}\mathbf{1}$ for a pair of solutions to some equation $\mathbf{y}\text{'}\text{'}\mathbf{+}\mathbf{py}\text{'}\mathbf{+}\mathbf{qy}\mathbf{=}\mathbf{0}$ (with $\mathbf{p}$ and $\mathbf{q}$ continuous).

(a) $\mathbf{w}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{=}\mathbf{6}{\mathbf{e}}^{\mathbf{4}\mathbf{t}}$

(b) $\mathbf{w}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{=}{\mathbf{t}}^{\mathbf{3}}$

(c) $\mathbf{w}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{=}\mathbf{(}\mathbf{t}\mathbf{+}\mathbf{1}{\mathbf{)}}^{\mathbf{-}\mathbf{1}}$

(d) $\mathbf{w}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{\equiv }\mathbf{0}$

Step-by-Step Solution

Answer

(a) The given function $\mathrm{w}\left(\mathrm{t}\right)=6{\mathrm{e}}^{4\mathrm{t}}$ is Wronskian.

(b) Given function $\mathrm{w}\left(\mathrm{t}\right)={\mathrm{t}}^3$ is not a Wronskian.

(c) The given function $\mathrm{w}\left(\mathrm{t}\right)=(\mathrm{t}+1{)}^{-1}$ is Wronskian.

(d) Given function $\mathrm{w}\left(\mathrm{t}\right)\equiv 0$ is not a Wronskian.

Step 1: Check whether the given function is Wronskian or not

Given interval is $-1<\mathrm{t}<1$. The given function can be Wronskian if the function is not equal to zero in the given interval.

 

Given function is $\mathrm{w}\left(\mathrm{t}\right)=6{\mathrm{e}}^{4\mathrm{t}}$

 

This function is always positive and cannot be zero in the interval $-1<\mathrm{t}<1$.Therefore, this function can be Wronskian.

Step 2: Check whether the given function is Wronskian or not

Given function is $\mathrm{w}\left(\mathrm{t}\right)={\mathrm{t}}^3$

 

This function can be positive and negative and can be equal to zero in the interval $-1<\mathrm{t}<1$.Therefore, this function cannot be Wronskian.

Step 3: Check whether the given function is Wronskian or not

Given function is $\mathrm{w}\left(\mathrm{t}\right)=(\mathrm{t}+1{)}^{-1}$

 

This function can be positive and negative and cannot be equal to zero in the interval $-1<\mathrm{t}<1$.

 

Therefore, this function can be Wronskian.

Step 4: Check whether the given function is Wronskian or not

Given function is $\mathrm{w}\left(\mathrm{t}\right)\equiv 0$

This function is always zero in the interval $\mathbf{-}\mathbf{1}\mathbf{<}\mathbf{t}\mathbf{<}\mathbf{1}$.Therefore, this function cannot be Wronskian.

The given function can be Wronskian if the function is not equal to zero in the given interval.