Question

Let $y_{1}=e^{-x} cosx, y_{2}=e^{-x} sinx$. Choose the correct statements out of the following:
A. the Wronskian determinant is zero
B. the Wronskian determinant is $e^{-2x}$}
C. $y_{1}$ and $y_{2}$ are linearly dependent}
D. $y_{1}$ and $y_{2}$ are linearly independent} Choose the correct answer from the options given below:

• A. A, C only
• B. A, D only
• C. B, C only
• D. B, D only

Hint:

For functions of the form $e^{\alpha x}\cos \beta x$ and $e^{\alpha x}\sin \beta x$, the Wronskian is always $\beta e^{2\alpha x}$. Here $\alpha = -1, \beta = 1$, so $W = 1 \cdot e^{-2x}$.

Answer 1

The Correct Option is D