Q33E - Fundamentals Of Differential Equations And Boundary Value Problems

Question

Explain why two functions are linearly dependent on an interval I if and only if there exist constants c₁ and c₂, not both zero, such that for ${\mathit{c}}_{\mathbf{1}}{\mathit{y}}_{\mathbf{1}}\mathbf{\left(}\mathit{t}\mathbf{\right)}\mathbf{+}{\mathit{c}}_{\mathbf{2}}{\mathit{y}}_{\mathbf{2}}\mathbf{\left(}\mathit{t}\mathbf{\right)}\mathbf{=}\mathbf{0}$ all t in I.

Step-by-Step Solution

Answer

The functions are linearly dependent.

Step 1: Apply the given values.

Here $c_1{y}_1\left(t\right)+c_2{y}_2\left(t\right)=0$         

 

If the y₁ and y₂ are linearly dependent then;

 $y_1\left(t\right)=C_1{y}_2\left(t\right)$

Step 2: Find linearly dependent functions

Now, 

$y_1\left(t\right)-C_1{y}_2\left(t\right)=0$ 

Let $C_1=-\frac{c_2}{c_1}$ then

 $$\begin{gathered} y_1\left(t\right)+\frac{c_2}{c_1}{y}_2\left(t\right)=0 \\ {c}_1{y}_1\left(t\right)+c_2{y}_2\left(t\right)=0 \end{gathered}$$         

 

Let $c_1=0 \text{and} c_2\ne 0$ then $y_2\left(t\right)=0$.

 

And if $c_2=0 \text{\hspace{0.05em}and} c_1\ne 0$ then $y_1\left(t\right)=0$

So, if both are not zero then both are linearly dependent.

 

Therefore, functions are linearly dependent.