COMEDK · Maths · 32. Differential Equations
The solution of \(dy = \cos x (2 - y \ \text{cosec} \ x) dx\) where \(y = \sqrt{2}\) when \(x = \dfrac{\pi}{4}\) is
- A \(y = \tan\left(\dfrac{x}{2}\right) + \cot\left(\dfrac{x}{2}\right)\)
- B \(y = \sin x + \dfrac{1}{2} \ \text{cosec} \ x\)
- C \(y \sin x = \dfrac{1}{2} \cos 2x\)
- D \(y = \dfrac{1}{\sqrt{2}} \sec x + \sqrt{2} \cos\left(\dfrac{x}{2}\right)\)
Answer & Solution
Correct Answer
(B) \(y = \sin x + \dfrac{1}{2} \ \text{cosec} \ x\)
Step-by-step Solution
Detailed explanation
The given differential equation can be rewritten as: \(\dfrac{dy}{dx} = 2\cos x - y \cos x \text{cosec} x\) \(\dfrac{dy}{dx} + y \cot x = 2\cos x\) This is a linear differential equation of the form \(\dfrac{dy}{dx} + P y = Q\), where \(P = \cot x\) and \(Q = 2\cos x\). The…
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