Q31E - Fundamentals Of Differential Equations And Boundary Value Problems

Question

In Problems 27–32, use Definition 1 to determine whether the functions y₁ and y₂ are linearly dependent on the interval (0, 1).

31. y₁(t) = tan²t - sec²t,y₂(t) = 3

Step-by-Step Solution

Answer

The functions are linearly dependent.

Step 1: Apply the given values.

The given functions are y₁(t) = tan²t - sec²t,y₂(t) = 3

If the y₁ and y₂ are linearly dependent on the interval (0, 1) then 

y₁(t) = C₁y₂(t)

Step 2: Find linearly dependent functions.

Check whether the function can be written as the constant multiple of others.

$\begin{array}{rcl}{y}_1\left(t\right)& =& C{y}_2\left(t\right)\\ {\tan }^{2}t - se{c}^{2}t& =& 3C\\ -1& =& 3C\\ C& =& -\frac{1}{3}\end{array}$

Since the value ofis a constant. 

Therefore, these functions are linear dependent.