Q30E - 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).

30. y₁(t) = t²cos(lnt), y₂(t) = t²sin(lnt)

Step-by-Step Solution

Answer

The functions are linearly independent.

Step 1: Apply the given values.

The given functions are y₁(t) = t²cos(lnt), y₂(t) = t²sin(lnt)

 

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.

$\begin{array}{rcl}{y}_1\left(t\right)& =& C{y}_2\left(t\right)\\ t^2\cos \left(\ln t\right)& =& C{t}^2\sin \left(\ln t\right)\\ C& =& cot\left(\ln t\right)\end{array}$

Sinceis cot(lnt) a non-constant function of t.

 

Therefore, these functions are linear independent.