Q29E - 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).

29. y₁(t) = te^2t, y₂(t) = e^2t

Step-by-Step Solution

Answer

The functions are linearly independent.

Step 1: Apply the given values.

The functions are y₁(t) = te^2t, y₂(t) = e^2t

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

y₁(t) = C₁y₂(t)

Step 2: Find linearly dependent functions.

Now, 

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

Since the function can be written as constant multiple of others, therefore they are linearly independent.

 

Thus, these functions are linear independent.