MAT244--2019F > Quiz-3

TUT0402 question

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shangluy:
Find the Wronskian of $cos^2\theta$ and $1 + cos2\theta$

Let $f = cos^2\theta$ and $g = 1 + cos2\theta$, then
\begin{equation}
    f' = -2sin\theta cos\theta = -sin2\theta \quad\quad\quad g' = -2sin2\theta
\end{equation}

since $W = fg' - f'g$, we get
\begin{align*}
    W &= cos^2\theta(-2sin2\theta) - (-sin2\theta)(1 + cos2\theta)\\
    &= -2cos^2\theta sin2\theta + sin2\theta(1 + cos2\theta)\\
    &= sin2\theta(-2cos^2\theta + 1 + cos2\theta)\quad\quad\quad\quad by \quad(2cos^2\theta - 1 = cos2\theta)\\
    &= sin2\theta(-cos2\theta + cos2\theta)\\
    &= 0
\end{align*}
Therefore the Wronskian of $cos^2\theta$ and $1 + cos2\theta$ is $0$

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