If \(\sqrt{r}=a e^{\theta \cot \alpha}\) where \(a\) and \(\alpha\) are real numbers, then \(\frac{d^{2} r}{d \theta^{2}}-4 r \cot ^{2} \alpha\) is

Given, \(\quad \sqrt{r}=a e^{\theta \cot \alpha}\)
Differentiating w.r.t. \(\theta\),
\[
\begin{aligned}
\frac{1}{2 \sqrt{r}} \frac{d r}{d \theta} &=a \cot \alpha \cdot e^{\theta \cot \alpha} \\
\frac{d r}{d \theta} &=2 a \sqrt{r} \cot \alpha e^{\theta \cot \alpha} \\
\frac{d r}{d \theta} &=2 a \cdot a e^{\theta \cot \alpha} \cdot \cot \alpha \cdot e^{\theta \cot \alpha}
\end{aligned}
\]
[from Eq. (i)]
\[
\frac{d r}{d \theta}=2 a^{2} \cot \alpha \cdot \alpha \cdot e^{2000 t \alpha}
\]
Again \(r\) differentiating w.r.t. \(\theta\)
\(\frac{d^{2} r}{d \theta^{2}}=2 a^{2} \cot \alpha \cdot e^{2 \theta \cot \alpha} \cdot 2 \cot \alpha\) \(\frac{d^{2} r}{d \theta^{2}}=4 a^{2} \cot ^{2} \alpha \cdot e^{20 \cot \theta}\) \(\frac{d^{2} r}{d \theta^{2}}=4 \cot ^{2} \alpha \cdot\left(a e^{\theta \cot \alpha}\right)^{2}\) \(\frac{d^{2} r}{d \theta^{2}}=4 \cot ^{2} \alpha \cdot(\sqrt{r})^{2} \quad\) [from Eq. (i)] \(\frac{d^{2} r}{d \theta^{2}}-4 r \cot ^{2} \alpha=0\)