Q32E - 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).

32. y₁(t)= 0, y₂(t) = e^t

Step-by-Step Solution

Answer

The functions are linearly dependent.

Step 1: Apply the given values.

The functions are y₁(t)= 0, y₂(t) = e^t

 

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,

y₁(t) = Cy₂(t)

   0 = Ce^t

   C = 0 

The value of C is a constant. 

Therefore, these functions are linear dependent.