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TS EAMCET · Maths · Differential Equations

The general solution of the differential equation \(\frac{d y}{d x}+\frac{\sin (2 x+y)}{\cos x}+2=0\)

  1. A \((\sec x+\tan x)[\operatorname{cosec}(2 x+y)-\cot (2 x+y)]=c\)
  2. B \(\sin (2 x+y) \cos x=c\)
  3. C \(\cos (2 x+y) \sin x=c\)
  4. D \((\operatorname{cosec} x-\cot x)(\sec (2 x+y)-\tan (2 x+y))=c\)
Verified Solution

Answer & Solution

Correct Answer

(A) \((\sec x+\tan x)[\operatorname{cosec}(2 x+y)-\cot (2 x+y)]=c\)

Step-by-step Solution

Detailed explanation

\(\frac{d y}{d x}+\frac{\sin (2 x+y)}{\cos x}+2=0\)....(i) Let \(2 x+y=t\) \(\Rightarrow 2+\frac{d y}{d x}=\frac{d t}{d x}\). Then, \(\frac{d t}{d x}+\frac{\sin t}{\cos x}=0 \quad[\) from(i)] \(\Rightarrow-\operatorname{cosec} t d t=\sec x d x\)…