Q28E - 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).

28. y₁(t) = e^3t, y₂(t) = e^-4t

Step-by-Step Solution

Answer

The functions are not linearly dependent.

Step 1: Apply the given values.

The given functions are y₁(t) = e^3t, y₂(t) = e^-4t

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,

${\mathit{e}}^{\mathbf{3}\mathbf{t}}\mathbf{\ne }\mathit{C}{\mathit{e}}^{\mathbf{-}\mathbf{4}\mathbf{t}}$

This cannot be expressed as e^3t multiples of e^-4t.

Therefore, these functions are not linear dependent.