Differential Equations Question 273
Question: The function $ f(\theta )=\frac{d}{d\theta }\int\limits_0^{\theta }{\frac{dx}{1-\cos \theta \cos x}} $ satisfies the differential equation
Options:
A) $ \frac{df}{d\theta }+2f(\theta )cot\theta =0 $
B) $ \frac{df}{d\theta }-2f(\theta )cot\theta =0 $
C) $ \frac{df}{d\theta }+2f(\theta )=0 $
D) $ \frac{df}{d\theta }-2f(\theta )=0 $
Show Answer
Answer:
Correct Answer: A
Solution:
[a] we have $ f(\theta )=\frac{d}{d\theta }\int\limits_0^{\theta }{\frac{dx}{1-\cos \theta \cos x}} $
$ =\frac{1}{1-{{\cos }^{2}}\theta }=\cos ec^{2}\theta $ (using Leibnitz’s Rule)
$ \Rightarrow \frac{df(\theta )}{d\theta }=-2\cos ec^{2}\theta \cot \theta $
$ \Rightarrow \frac{df(\theta )}{d\theta }+2f(\theta )cot\theta =0 $
BETA
