Q27E - 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).

27. y₁(t) = costsint, y₂(t) = sin2t

Step-by-Step Solution

Answer

The functions are linearly dependent y₁(t)=2Cy₂ for any C.

Step 1: Apply the given values.

Here y₁(t) = costsint, y₂(t) = sin2t 

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

y₁(t) = C₁y₂(t)

The trigonometric identity is $\sin 2\theta =2\sin \theta \cos \theta$.

Step 2: Find linearly dependent functions.

Now,

$\begin{array}{rcl}{y}_1\left(t\right)& =& c_1{y}_2\left(t\right)\\ \cos \left(t\right)\sin \left(t\right)& =& 2C_1\sin \left(t\right)\cos \left(t\right)\end{array}$

Therefore, y₁(t) = 2Cy₂(t), where C = 2C₁,

 

Thus, these functions are linear dependent.